11 Discrimination of Quantum States · than two states it is diﬃcult to treat mixed states. The...

11 Discrimination of Quantum States

Janos A. Bergou1, Ulrike Herzog2, and Mark Hillery1

1 Department of Physics and Astronomy, Hunter College of The City Universityof New York, 695 Park Avenue, New York, NY 10021,[email protected], [email protected]

2 Institut fur Physik, Humboldt-Universitat Berlin, Newtonstrasse 15,12489 Berlin, Germany, [email protected]

Abstract. The problem of discriminating among given nonorthogonal quantumstates is underlying many of the schemes that have been suggested for quantumcommunication and quantum computing. However, quantum mechanics puts severelimitations on our ability to determine the state of a quantum system. In particu-lar, nonorthogonal states cannot be discriminated perfectly, even if they are known,and various strategies for optimum discrimination with respect to some appropri-ately chosen criteria have been developed. In this article we review recent theoreti-cal progress regarding the two most important optimum discrimination strategies.We also give a detailed introduction with emphasis on the relevant concepts ofthe quantum theory of measurement. After a brief introduction into the field, thesecond chapter deals with optimum unambiguous, i. e error-free, discrimination.Ambiguous discrimination with minimum error is the subject of the third chap-ter. The fourth chapter is devoted to an overview of the recently emerging subfieldof discriminating multiparticle states. We conclude with a brief outlook where weattempt to outline directions of research for the immediate future.

11.1 Introduction

In quantum information and quantum computing the carrier of informationis some quantum system and information is encoded in its state [1]. Thestate, however, is not an observable in quantum mechanics [2] and, thus,a fundamental problem arises: after processing the information - i.e. afterthe desired transformation is performed on the input state by the quan-tum processor - the information has to be read out or, in other words, thestate of the system has to be determined. When the possible target statesare orthogonal, this is a relatively simple task if the set of possible states isknown. But when the possible target states are not orthogonal they cannotbe discriminated perfectly, and optimum discrimination with respect to someappropriately chosen criteria is far from being trivial even if the set of thepossible nonorthogonal states is known. Thus the problem of discriminat-ing among nonorthogonal states is ubiquitous in quantum information andquantum computing, underlying many of the communication and computingschemes that have been suggested so far. It is the purpose of this article toreview various theoretical schemes that have been developed for discriminat-ing among nonorthogonal quantum states. The corresponding experimental

J.A. Bergou, U. Herzog, M. Hillery, Discrimination of Quantum States, Lect. Notes Phys. 649,417–465 (2004)http://www.springerlink.com/ c© Springer-Verlag Berlin Heidelberg 2004

418 Janos A. Bergou, Ulrike Herzog, and Mark Hillery

realizations will be mentioned only in a cursory manner as they are reviewedelsewhere in this volume by Chefles.

The field of discriminating among nonorthogonal quantum states has beenaround for quite some time now [3]. Stimulated by the rapid developmentsin quantum information theory of the 90’s the question of how to discrimi-nate between nonorthogonal quantum states in an optimum way has gainedrenewed interest. The developments until about the late 90’s are reviewed inan excellent review article by Chefles [4]. Therefore, in this review we willmainly focus our attention to recent advances not contained in [4] and willcite earlier results only when they are necessary for the understanding of thenewer ones.

In order to devise an optimum state-discriminating measurement, strate-gies have been developed with respect to various criteria. In this review ar-ticle we restrict ourselves to the two most obvious criteria for optimizinga measurement scheme that is designed for discriminating between differ-ent states of a quantum system, being either pure states or mixed states.The two methods are optimum unambiguous discrimination of the states, onthe one hand, and state discrimination with minimum error, on the otherhand. They will be outlined in detail in the next two sections of this re-view. In particular, at the beginning of these sections we give a tutorialintroduction to the two main strategies by considering simple examples,namely unambiguous discrimination of two pure states in Sect. 11.2, andminimum-error discrimination of two mixed states in Sect. 11.3. This willallow us to introduce the concept of generalized measurements along withthe other typical theoretical tools employed in problems of this sort. Se-lected applications of the unambiguous discrimination strategy, one chosenfrom quantum communication and the other from quantum computing willbe reviewed at the end of Sect. 11.2. Section 11.4 is devoted to the recentlyemerging sub-field of discriminating among multipartite states by means oflocal operations and classical communication (LOCC). We conclude witha brief outlook in Sect. 11.5. Throughout the article we assume that foreach measurement only a single copy of the quantum system is available.However, the case of multiple copies could be easily accounted for, in themeasurement schemes we consider, if the states are replaced by their corre-sponding multi-fold tensor products. We note that apart from the two op-timization schemes we consider, state-distinguishing measurements can alsobe optimized with respect to other criteria, such as requiring a maximumof the mutual information [5] or of the fidelity [6]. In particular, the for-mer approach is of importance for the transmission of quantum informa-tion. From a mathematical point of view, however, these other criteria posemuch bigger problems, since they rely on optimizing a nonlinear functionalof the given states and, therefore, are beyond the scope of the current re-view.

11 Discrimination of Quantum States 419

11.2 Unambiguous Discrimination

In this section we will review schemes for unambiguous discrimination. Al-though the simplest case of two pure states is well known and has beenreviewed extensively (see, for example, [4]), from a pedagogical point of viewwe find it useful to include it here as well, because many of the techniquesemployed later can be best understood on this simple example. We also wantto mention that the two main discrimination strategies evolved rather differ-ently from the very beginning. On the one hand, unambiguous discriminationstarted with pure states and only very recently turned its attention to discrim-inating among mixed quantum states. At the end of this section we will reviewrecent progress in this area. On the other hand, minimum-error discrimina-tion addressed the problem of discriminating among two mixed quantumstates from the very beginning and the results for two pure states followed asspecial cases. Each strategy has its own advantages and drawbacks. While un-ambiguous discrimination is relatively straightforward to generalize for morethan two states it is difficult to treat mixed states. The error-minimizingapproach, initially developed for two mixed states, is hard to generalize formore than two states.

Before we begin our systematic study of the optimal unambiguous dis-crimination of two pure states in the next subsection, we want to gain somephysical insight first. To this end, let us describe the procedure of optimumunambiguous discrimination between two nonorthogonal single photon po-larization states in terms of classical optics. We consider a series of weakpulses containing, on the average, much less than one photon. Each pulseis linearly polarized, with equal probability, either in the direction e1 orin the direction e2 with e1,2 = cosΘ ex ± sinΘ ey, where we assume thatcosΘ ≥ sinΘ. If the pulses pass through a linear optical device and undergopolarization-selective linear attenuation in the x direction, their polarizationvectors can be made orthogonal. For this purpose the attenuation processhas to be designed in such a way that the amplitude of the x-componentis reduced by a factor of tanΘ. Due to the attenuation, the initial electricfield vectors E1,2 = E0e1,2 of the respective pulses are then transformed intothe vectors E′

1,2 = E0√

2 sinΘ(ex ± ey)/√

2. Hence, after leaving the linearoptical device, the polarization directions of the two kinds of pulses are or-thogonal and, therefore, can be discriminated unambiguously even when thepulses contain only a single photon. However, the intensity that can be usedfor unambiguous polarization state discrimination is reduced by a factor of2 sin2Θ relative to the total initial intensity. Since a classical intensity ratiocorresponds to the probability ratio of detecting a photon, every incomingphoton will yield an unambiguous result in the state discrimination processonly with the probability

PD = 2 sin2Θ = 1− cos(2Θ) = 1− |e1e2| . (11.1)


Here we introduced the modulus of the scalar product since, when the direc-tion of one of the vectors, e1 or e2, is reversed, the linear polarization stateremains the same. The probability that the discrimination procedure fails,i. e. that the polarization state of the photon cannot be determined unam-biguously, is therefore QF = 1 − PD = |e1e2|. Although orthogonalizationof the polarization states is also possible when polarization-selective attenu-ation affects a polarization direction that differs from the x-axis, it is easyto see that the symmetric procedure described above is optimal in the sensethat it yields the maximum achievable value of PD, or the minimum value ofQF = 1− PD, respectively.

From the pioneering investigations of unambiguous discrimination be-tween general nonorthogonal quantum states it follows that for any two states,|ψ1〉 and |ψ2〉, occurring with equal a priori probability, the optimum proba-bility of obtaining an unambiguous result is given by (11.1) when the scalarproduct e1e2 is replaced by the overlap 〈ψ1|ψ2〉, as we will show in the fol-lowing [see (11.2), in particular].

11.2.1 Unambiguous Discrimination of Two Pure States

Unambiguous discrimination started with the work of Ivanovic [7] who stud-ied the following problem. A collection of quantum systems is prepared sothat each single system is equally likely to be prepared in one of two knownstates, |ψ1〉 or ψ2〉. Furthermore, the states are not orthogonal, 〈ψ1|ψ2〉 �= 0.The preparer then hands the systems over to an observer one by one whosetask is to determine which one of the two states has actually been preparedin each case. All the observer can do is to perform a single measurement orperhaps a series of measurements on the individual system. Ivanovic came tothe conclusion that if one allows inconclusive detection results to occur thenin the remaining cases the observer can conclusively determine the state ofthe individual system.

It is rather easy to see that a simple von Neumann measurement can ac-complish this task. Let us denote the Hilbert space of the two given statesby H and introduce the projector P1 for |ψ1〉 and P1 for the orthogonal sub-space, such that P1 + P1 = 1, the unity in H. Then we know for sure that|ψ2〉 was prepared if in the measurement of {P1, P1} a click in the P1 detectoroccurs. A similar conclusion for |ψ1〉 can be reached with the roles of |ψ1〉 and|ψ2〉 reversed. Of course, when a click along P1 (or P2) occurs then we learnnothing about which state was prepared thus corresponding to inconclusiveresults. Ivanovic’s startling observation was that a sequence of measurementscan sometimes do better than a single von Neumann measurement describedhere. Dieks [8] then found that this sequence of measurements can be real-ized with a single generalized measurement (POVM) and Peres subsequentlyshowed that this POVM is optimal in the sense that its failure probability, theprobability that an inconclusive outcome occurs, is minimum [9]. The prob-ability of the inconclusive outcome, or failure, is QIDP = |〈ψ1|ψ2〉| and the


probability of success is then given by what is called the Ivanovic-Dieks-Peres(IDP) limit,

PIDP = 1−QIDP = 1− |〈ψ1|ψ2〉| . (11.2)

This result can be generalized for the case when the preparation probabilitiesof the states, η1 and η2, are different, η1 �= η2. The preparation probabilitiesare also called a priori probabilities. The IDP result thus corresponds tothe case of equal a priori probabilities for the two states, η1 = η2 = 1/2,and the generalization for arbitrary a priori probabilities is due to Jaegerand Shimony [10]. Here we briefly review the derivation of the general resultfollowing [10] and the very readable account by Ban [11].

11.2.2 Optimal POVM and the Complete Solution

The von Neumann projective measurement described above has two out-comes. It can correctly identify one of the two states at the expense of missingthe other completely and occasionally missing the identifiable one, as well.If we want to do better we would like to have a measurement with three- not just two - outcomes, |ψ1〉, |ψ2〉 and failure. In the two-dimensionalHilbert space H the number of possible outcomes for a von Neumann mea-surement cannot exceed two, since it is always restricted by the dimensional-ity of the Hilbert space. We have to turn to generalized measurements thatallow greater flexibility than simple projective measurements [12]. In partic-ular, the number of distinguishable outcomes can exceed the dimensionalityof the corresponding Hilbert space. For our case this means that we replacethe projector P2 by the quantum detection operator Π1, P1 by Π2 and intro-duce Π0 for the inconclusive results in such a way that 〈ψ1|Π1|ψ1〉 = p1 isthe probability of successfully identifying |ψ1〉, 〈ψ1|Π0|ψ1〉 = q1 is the prob-ability of failing to identify |ψ1〉, (and similarly for |ψ2〉). For unambiguousdiscrimination we then require 〈ψ2|Π1|ψ2〉 = 〈ψ1|Π2|ψ1〉 = 0. We want thesepossibilities to be exhaustive,

Π1 +Π2 +Π0 = I , (11.3)

where I is the unity operator in H. The probabilities are always real and non-negative which implies that the quantum detection operators are Hermiteanand non-negative or, in other words, positive semi-definite.

Clearly, (11.3) does not correspond to orthogonal measurements when alldetection operators are different from zero. It describes an operation calledpositive operator-valued measure (POVM) or simply a generalized measure-ment with the detection operators as its elements.

We now want to determine these operators explicitly. Consider the oper-ator Ak = UkΠ

1/2k , where Uk is an arbitrary unitary operator (k = 0, 1, 2).

From this expression we immediately obtain Πk = A†kAk and the detection


probability can be expressed as 〈ψi|A†kAk|ψi〉 = ‖Akψi‖2 ≥ 0 where ‖...‖

stands for the norm. This expression also helps us to identify the so far arbi-trary operator Ak. The expression Ak|ψi〉 corresponds to the post-detectionstate. Because of the positivity of the norm, the condition of unambiguousdiscrimination is equivalent to the requirement

A1|ψ2〉 = A2|ψ1〉 = 0 . (11.4)

If we introduce |ψ⊥i 〉 as the vector orthogonal to |ψi′〉 (i �= i′) - a nota-

tion that will become obvious in Sect. 11.2.2 - then A1 = c1|ψ1〉〈ψ⊥1 | and

A2 = c2|ψ2〉〈ψ⊥2 |. Here ci are complex coefficients to be determined from the

condition of optimum and |ψ1〉 and |ψ2〉 are the post-detection states, nor-malized to unity. For perfect distinguishability of the post-detection states,corresponding to optimal discrimination, we have to require their orthogonal-ity, 〈ψ1|ψ2〉 = 0, so they can be represented by a pair of arbitrarily directedorthogonal vectors in H.

With the help of these expressions we can write the detection operatorsas Π1 = A†

1A1 = |c1|2|ψ⊥1 〉〈ψ⊥

1 | and Π2 = A†2A2 = |c2|2|ψ⊥

2 〉〈ψ⊥2 |. Inserting

these expressions in the definition of p1 and p2 gives |c1|2 = p1/|〈ψ1|ψ⊥1 〉|2

and a similar expression for |c2|2. Finally, introducing cosΘ = |〈ψ1|ψ2〉| andsinΘ = |〈ψ1|ψ⊥

1 〉|, we can write the detection operators as

Π1 =p1

sin2Θ|ψ⊥

1 〉〈ψ⊥1 | ,

Π2 =p2

sin2Θ|ψ⊥

2 〉〈ψ⊥2 | . (11.5)

Now, Π1 and Π2 are positive semi-definite operators by construction. How-ever, there is one additional condition for the existence of the POVM whichis the positivity of the inconclusive detection operator,

Π0 = I −Π1 −Π2 . (11.6)

This is a simple 2 by 2 matrix in H and the corresponding eigenvalue problemcan be solved analytically. Non-negativity of the eigenvalues leads, after sometedious but straightforward algebra, to the condition

q1q2 ≥ |〈ψ1|ψ2〉|2 , (11.7)

where q1 = 1 − p1 and q2 = 1 − p2 are the failure probabilities for thecorresponding input states.

Equation (11.7) represents the constraint imposed by the positivity re-quirement on the optimum detection operators. The task we set out to solvecan now be formulated as follows. Let

Q = η1q1 + η2q2 (11.8)

denote the average failure probability for unambiguous discrimination. Wewant to minimize this failure probability subject to the constraint, (11.7).


Due to the relation, P = η1p1 + η2p2 = 1−Q, the minimum of Q also givesus the maximum probability of success. Clearly, for optimum the productq1q2 should be at its minimum allowed by (11.7), and we can then expressq2 with the help of q1 as q2 = cos2Θ/q1. Inserting this expression in (11.8)yields

Q = η1q1 + η2cos2Θq1

, (11.9)

where q1 can now be regarded as the independent parameter of the problem.Optimization of Q with respect to q1 gives qPOVM1 =

√η2/η1 cosΘ and

qPOVM2 =√η1/η2 cosΘ. Finally, substituting these optimal values into (11.8)

gives the optimum failure probability,

QPOVM = 2√η1η2 cosΘ . (11.10)

For η1 = η2 = 1/2 this reproduces the IDP result, (11.2), as it should.Let us next see how this result compares to the average failure probabil-

ities of the two possible unambiguously discriminating von Neumann mea-surements that were described at the beginning of this section. The averagefailure probability for the first von Neumann measurement, with its failuredirection along |ψ1〉, can be written by simple inspection as

Q1 = η1 + η2|〈ψ1|ψ2〉|2 , (11.11)

since |ψ1〉 gives a click with probability 1 in this direction but it is onlyprepared with probability η1 and |ψ2〉 gives a click with probability |〈ψ1|ψ2〉|2but it is only prepared with probability η2. The corresponding detector set-up, yielding Q1 for the failure probability, is depicted in Fig. 11.1.

By entirely similar reasoning, the average failure probability for the secondvon Neumann measurement, with its failure direction along |ψ2〉, is given by

Q2 = η1|〈ψ1|ψ2〉|2 + η2 . (11.12)

The corresponding detector set-up, yielding Q2 for the failure probability, isdepicted in Fig. 11.2.

What we can observe is that Q1 and Q2 are given as the arithmetic meanof two terms and QPOVM is the geometric mean of the same two terms foreither case. So, one would be tempted to say that the POVM performs betteralways. This, however, is not quite the case, it does so only when it exists. Theobvious condition for the POVM solution to exists is that both qPOVM1 ≤ 1and qPOVM2 ≤ 1. Using η2 = 1− η1, a little algebra tells us that the POVMexists in the range cos2Θ/(1 + cos2Θ) ≤ η1 ≤ 1/(1 + cos2Θ). If η1 is smallerthan the lower boundary, the POVM goes over to the first von Neumannmeasurement and if η1 exceeds the upper boundary the POVM goes over tothe second von Neumann measurement. This can be easily seen from (11.5)and (11.6) since p1 = 1−q1 = 0 for q1 = 1 and Π0 becomes a projection along


Fig. 11.1. A von Neumann measurement that discriminates |ψ2〉 unambiguously.The detector D0 = P1 is set up along |ψ1〉 and the detector D2 = P1 is set upalong the orthogonal direction. When a click in the D2 detector occurs we know forcertain that |ψ2〉 was prepared as the input state since it is the only one that hasa component along this direction. When a click in the detector D0 occurs we learnnothing about which state was prepared as the input since both have a componentalong this direction. This measurement outcome then corresponds to the inconclu-sive result so D0 is the failure detector and the probability that it clicks is Q1.

Fig. 11.2. A von Neumann measurement that discriminates |ψ1〉 unambiguously.The detector D0 = P2 is set up along |ψ2〉 and the detector D1 = P2 is set upalong the orthogonal direction. When a click in the D1 detector occurs we know forcertain that |ψ1〉 was prepared as the input state since it is the only one that hasa component along this direction. When a click in the detector D0 occurs we learnnothing about which state was prepared as the input since both have a componentalong this direction. This measurement outcome then corresponds to the inconclu-sive result so D0 is the failure detector and the probability that it clicks is Q2.


Fig. 11.3. Optimal POVM that discriminates |ψ1〉 and |ψ2〉 unambiguously. Thedetector D0 = Π0 is set up symmetrically between |ψ1〉 and |ψ2〉, the detector D1 =Π1 is set up along P2 and the detector D2 = Π2 is set up along P1. When a click inthe Di detector occurs we know for certain that |ψi〉 was prepared (i = 1, 2) as theinput state since it is the only one that has a component along this direction. Whena click in the detector D0 occurs we learn nothing about which state was preparedas the input since both have a component along this direction. This measurementoutcome then corresponds to the inconclusive result so D0 is the failure detectorand the probability that it clicks is QPOV M . (The figure is for illustrative purposesonly. Arrows representing POVM detection operators do not correspond to simpleprojections along their respective directions. They are drawn shorter than arrowsrepresenting state vectors which, in turn, have unit length. For an implementationof the POVM, see Sect. 11.2.3.)

|ψ1〉 (and correspondingly for p2 = 0). The set-up of the detection operators,yielding QPOVM for the failure probability, is depicted in Fig. 11.3.

These findings can be summarized as follows. The optimal failure proba-bility, Qopt, is given as

Qopt =

QPOVM if cos2 Θ

1+cos2 Θ ≤ η1 ≤1

1+cos2 Θ ,

Q1 if η1 < cos2 Θ1+cos2 Θ ,

Q2 if 11+cos2 Θ < η1 .

(11.13)

Figure 11.4 displays the failure probabilities vs. η1 for a fixed value of theoverlap, cos2Θ.

The above result is very satisfying from a physical point of view. ThePOVM delivers a lower failure probability in its entire range of existencethan either of the two von Neumann measurements. At the boundaries ofthis range it merges smoothly with the von Neumann measurement thathas a lower failure probability at that point. Outside this range the statepreparation is dominated by one of the states and the optimal measurementbecomes a von Neumann projective measurement, using the state that isprepared less frequently as its failure direction.

Next we turn our attention to the physical implementation of the POVM.


0 0.2 0.4 0.6 0.8 1Η1

0.2

0.4

0.6

0.8

1

Q

Fig. 11.4. Failure probability, Q, vs. the prior probability, η1. Dashed line: Q1,dotted line: Q2, solid line: QPOV M . For the figure we used the following represen-tative value: |〈ψ1|ψ2〉|2 = 0.1. For this the optimal failure probability, Qopt is givenby Q1 for 0 < η1 < 0.09, by QPOV M for 0.09 ≤ η1 ≤ 0.9 and by Q2 for 0.9 < η1.

11.2.3 Neumark’s Theorem and the Realization of the POVM

The problem with the POVM is that it is very hard to realize in the orig-inal system Hilbert space since it involves detection operators that do notcorrespond to orthogonal projectors. In fact, they are not projectors at all.Fortunately, there is an essential result, known as Neumark’s theorem [13]. Itstates that a POVM can be realized by the following constructive procedure,also known as generalized measurement. The system is embedded in a largerHilbert space where the extra degrees of freedom are customarily called theancilla. Then a unitary transformation entangles the system degrees of free-dom with those of the ancilla. Finally, after this interaction, projective vonNeumann measurements can be carried out on this larger system. As a con-sequence of the entanglement between the original system and the ancilla, aprojective measurement on the larger system will also transform the systemstate in the original Hilbert space. One can choose the unitary transformationand the subsequent von Neumann measurement in such a way that if outcomek is found in the von Neumann measurement on the larger system, the result-ing transformation on the original system state is Ak|ψ〉, i. e. it correspondsto an element of the POVM in the original system Hilbert space.

We illustrate the power of this theorem on an alternative derivation ofthe condition on the individual failure probabilities, (11.7). The joint Hilbertspace K of the ‘system plus ancilla’ is a tensor product of the two Hilbert


spaces, H of the system and A of the ancilla, K = H ⊗A. This means thata state in K is a superposition of product states where, in each product, thefirst member is from H and the second is from A. Specifically, the two inputsnow correspond to |ψ1〉|φ0〉 and |ψ2〉|φ0〉, where |φ0〉 describes the initial stateof the ancilla. We choose the unitary transformation as

U(|ψ1〉|φ0〉) =√p1|ψ′

1〉|φ0〉+√q1|ψ0〉|φ1〉 , (11.14)

U(|ψ2〉|φ0〉) =√p2|ψ′

2〉|φ0〉+√q2e

iθ|ψ0〉|φ1〉 , (11.15)

where |φ1〉 is chosen to be orthogonal to |φ0〉, and |ψ′1〉 and |ψ′

1〉 correspondto orthogonal vectors in the original Hilbert space. If we now perform a vonNeumann measurement on the ancilla then a click along the |φ1〉 directioncollapses both inputs onto the same output, |ψ0〉, and all information aboutthe inputs is lost. The probability that this happens for input i is qi (i =1, 2). Obviously, this outcome corresponds to the inconclusive result so qi arethe failure probabilities of the corresponding input states i. On the otherhand, a click along the |φ0〉 direction transforms the original inputs intoorthogonal outputs in the system Hilbert space. The probability that thishappens for input i is pi (i = 1, 2). Obviously, this outcome correspondsto full distinguishability in the original system Hilbert space, so pi are theprobabilities of success for discriminating the corresponding input states i.From unitarity we obtain pi + qi = 1 for i = 1, 2 and by taking the innerproduct of (11.14) and (11.15) we obtain (11.7). Thus Neumark’s theoremdelivers the positivity condition in a few lines and from here the rest ofthe derivation of the optimal failure probability goes along the same linesas for the POVM method. An optical implementation, based on the tensorproduct extension of the Hilbert space and employing linear optical elements(beam splitters and phase shifters) only, has been proposed in [14]. The statesto be discriminated are represented by a single photon that can be in asuperposition between two input ports of a six port interferometer with threeinput and three output ports. The third input corresponds to the ancilla,initially in the vacuum state. The desired unitary transformation is carriedout by this six port constructed from the appropriate linear optical elements.At the output side detectors are placed in front of each output port. If thephoton emerges from the ancilla port 3, the measurement failed. We canconstruct the device so that if the photon emerges from port i (i = 1, 2,and we know that the input state was i. Thus, detector clicks in the firsttwo output ports correspond to successful measurements. Measurements ofthis type can easily be generalized higher dimensional systems, using moregeneral multiports. An explicit example will be given in Sect. 3.2.

Before moving on to more uncharted territory we should note that theexample of unambiguous discrimination between two nonorthogonal polariza-tion states by using a polarization-sensitive absorber that was considered atthe beginning of this section does not conform to the scheme described here,although it clearly accomplishes its goal. Rather, it corresponds to another


kind of possible extension of the original system Hilbert space via the directsum method. In this case we append the ancilla space, A, to the system spaceH to form the joint Hilbert space K of the ‘system plus ancilla’. K is now adirect sum, K = H⊕A, of H and A, meaning that a state in K is a superpo-sition of two terms where the first one lies entirely in H and the second onein A. In particular, the two input states, |ψ1〉 and |ψ2〉, have no componentsin the ancilla space. In the example with the two single photon polarizationstates spanning the two-dimensional system Hilbert space this means thatwe simply include a third state, the vacuum, as the ancilla space. We cansay that our two-dimensional objects, the qubits, secretly live in the three-dimensional space of qutrits. The described unambiguous discrimination pro-cedure between two single-photon polarization states has been successfullyperformed experimentally, implementing the polarization-selective attenua-tion either by polarization-dependent absorption in a fiber [15] or with thehelp of polarizing beamsplitters [16]. The transformation corresponding tothe linear absorber redistributes the population among the three basis statesin such a way that the parts of the input qubit states that remain in the orig-inal qubit Hilbert space become orthogonal there. In this method we do notkeep track of what happens to the photon in the absorption process, whereasin the tensor product method, including a complete description of the ancilla(environment), we do. For the simple examples that we consider, one canintroduce equivalent notations for the two alternative methods. Therefore, inthe next section we will employ the somewhat simpler notations of the directsum method.

This completes our tutorial review of the unambiguous discriminationof two pure states and illustrates the two possible approaches, the directPOVM method and the method of generalized measurements based on Neu-mark’s theorem. We now turn our attention to more complicated problemsand briefly review recent progress in dealing with them.

11.2.4 More than Two Pure States

So far we have only considered discriminating between two states, and wehave seen that in that case a complete solution can be given. For more thantwo states, however, there are only a few general results, and explicit solutionsexist only for special cases. Here we shall review what is known.

Two general results apply to the case of unambiguous discrimination.The first, due to Chefles, is that only linearly independent states can beunambiguously discriminated [17]. This can be seen as follows. Let the POVMfor discriminating the N states |ψ1〉, . . . |ψN 〉 be given by the operators A1,. . . AN , and AI , where the operators act on the vectors in the space H, whichis the span of the vectors |ψ1〉, . . . |ψN 〉, and

A†IAI +

N∑j=1

A†jAj = I , (11.16)


which is an obvious generalization of (11.3) to N states. The operator AIagain corresponds to the inconclusive outcome, and the operator Aj , forj = 1 . . . N corresponds to identifying the state as |ψj〉. Because there mustbe no errors, we must have

〈ψk|A†jAj |ψk〉 = pjδjk, (11.17)

where 0 ≤ pj ≤ 1 is the probability of successfully identifying |ψj〉. Nowsuppose that the states are linearly dependent so that they can be expressedin terms of each other

|ψj〉 =N∑k=1

cjk|ψk〉. (11.18)

Substituting this into the above equation we have that

N∑m,n=1

c∗kmckn〈ψm|A†jAj |ψn〉 = pjδjk. (11.19)

This can be simplified by noting that

|〈ψm|A†jAj |ψn〉|2 ≤ 〈ψm|A

†jAj |ψm〉〈ψn|A

†jAj |ψn〉, (11.20)

which gives us that

〈ψm|A†jAj |ψn〉 = pjδmnδjm. (11.21)

Substituting this into (11.19) we find that |ckj |2 = δjk, which implies thatthe states are not linear combinations of each other and are, hence, linearlyindependent.

It is also possible to quickly draw some conclusions about the form of theoperator Aj . Because we have that

Aj |ψk〉 = 0, (11.22)

for j �= k, it annihilates the subspace Hk, which is the span of the vectors|ψ1〉, . . . |ψN 〉 with |ψk〉 omitted. Let |ψ⊥

k 〉 be the unit vector orthogonalto Hk which, by the way, explains the notation introduced in the previoussection. We can then choose

Aj =√pj

〈ψ⊥j |ψj〉

|ψ′j〉〈ψ⊥

j | , (11.23)

where |ψ′j〉, j = 1, . . . N are arbitrary orthogonal unit vectors. The remaining

problem is to find the values of pj . Let us denote the a priori probabilityof the state |ψj〉 by ηj . The values of pj should be chosen to maximize the


average success probability, P , where

P =N∑j=1

ηjpj . (11.24)

In addition, they must be chosen so that the operator

A†IAI = I −

N∑j=1

A†jAj

= I −N∑j=1

pj |ψ⊥j 〉〈ψ⊥

j ||〈ψj |ψ⊥

j 〉|2, (11.25)

is positive. This is a nontrivial problem.The second general result states that there do exist upper and lower

bounds on the success probability. Here we shall just present the results;the interested reader can consult the original papers for the derivations. Anupper bound is given by [18]

P ≤ 1− 1N − 1

N∑j=1

N∑k=1,k �=j

√ηjηk|〈ψj |ψk〉|. (11.26)

Building on work by Duan and Guo [19], X. Sun, et al. derived a lowerbound [20]. Consider the N ×N matrix whose elements are 〈ψj |ψk〉, and letλN be the smallest eigenvalue of this matrix. They showed that P ≥ λN .

The problem of discriminating among three nonorthogonal states was firstconsidered by Peres and Terno [21]. They developed a geometric approachand applied it numerically to several examples. A different method was con-sidered by Duan and Guo [19] and Y. Sun and ourselves [22]. We consideredthe three vectors to be discriminated, |ψj〉, j = 1, 2, 3, to lie in the spaceH. Tothis a “failure” space, A, is appended so that the whole problem takes placein the space obtained by the direct sum extension, K = H⊕A. If the proce-dure fails, the vector |ψj〉 is mapped into a vector in the failure space, |φj〉,and if it succeeds it is mapped onto a vector in the original space, √pj |ψ′

j〉,where ‖ψ′

j‖ = 1, and 0 ≤ pj ≤ 1. The vectors |ψ′j〉 are mutually orthogonal,

so that they can be perfectly distinguished. Chefles showed that the set offailure vectors must be linearly dependent for the optimal procedure [17], sothat the dimension of A will be one or two. One way of understanding thisresult is that if the failure vectors were linearly independent, then we couldperform a further unambiguous state discrimination procedure on them and,with some probability, tell which state we were originally given. This wouldimply that the original procedure was not optimal. Therefore, the optimalprocedure produces linearly dependent failure vectors, which cannot be fur-ther discriminated.


Making this more explicit, we assume that there is a unitary operator, U ,acting on K, such that

U |ψj〉 =√pj |ψ′

j〉+ |φj〉 . (11.27)

It should be noted that, unlike in (11.15), the vector |φj〉 is not normalizedto unity. Instead, we use 〈φj |φj〉 = qj here. After U has been performed,we measure the projection operator onto the space H. If we obtain 1, theprocedure has succeeded, and we know what the input state was. If the inputwas |ψj〉, the procedure succeeds with probability pj . If we obtain 0, theprocedure has failed, and this happens with probability qj = 1− pj = ‖φj‖2,if the input state was |ψj〉. The above equation implies that

〈φj |φk〉 = 〈ψj |ψk〉 − δjkpj . (11.28)

Defining the matrix Cjk = 〈φj |φk〉, we see by its definition that it must bepositive definite. Therefore, the problem of finding the optimal unambigu-ous state discrimination procedure reduces to finding the values of pj thatoptimize the success probability

P =3∑j=1

ηjpj , (11.29)

subject to the constraint that the 3×3 matrix, whose elements are 〈ψj |ψk〉−δjkpj , is positive.

This can be solved in some special cases. We shall assume that all of the apriori probabilities are the same, so that they are all 1/3. If all of the overlapsare the same, i.e.

〈ψ1|ψ2〉 = 〈ψ1|ψ3〉 = 〈ψ2|ψ3〉 = s, (11.30)

where s is real and positive, then qj = s, for j = 1, 2, 3, and Q = 1 − P = sas well.

There is also an explicit solution if

〈ψ1|ψ2〉 = 〈ψ1|ψ3〉 = s1 ,

〈ψ2|ψ3〉 = s2 , (11.31)

where both s1 and s2 are real and positive. We first note that for a fixedvalue of s1 there is a restriction on how large s2 can be. The largest theangle between |ψ2〉 and |ψ3〉 can be is twice the angle between |ψ1〉 and |ψ2〉(this maximum is achieved when the vectors are coplanar). This implies thats2 ≥ 2s21 − 1. The solution to the state discrimination problem depends onwhether s1/s2 < 2 or not. If it is, we have

q1 =s21s2, q2 = q3 = s2 ,

Q =13

[s21s2

+ 2s2

]. (11.32)


Fig. 11.5. An optical eight-port. The beams are straight lines, a suitable beamsplitter is placed at each point where two beams intersect, phase shifters are at oneinput of each beam splitter and at each output.

If s1/s2 ≥ 2, we have

q1 = 2s1 , q2 = q3 = s1 + s2 ,

Q =23(2s1 − s2) . (11.33)

This approach lends itself naturally to an optical implementation [22].The states to be discriminated are represented by a single photon that canbe in one of three modes. Additional modes (one or two, depending on thedimension of the failure space) that represent the ancilla, are initially in thevacuum state. Figure 11.5 displays the simpler case, that of an eight port.In general, the unitary transformation is carried out by an optical N port,where N is either 8 or 10, depending on the number of vacuum ports needed.It should be noted that an N port is a linear optical device with N/2 inputsand N/2 outputs, that can be constructed from beam splitters, phase shifters,and mirrors. At the output detectors are placed to determine from which ofthe ports the photon emerges. If it emerges from one of the failure ports, themeasurement has failed, but if it emerges from one of the other three, weknow what the input state was.

In particular, we can construct the device so that the three output portsthat correspond to a successful measurement are numbered 1 through 3,and a photon emerging from port j means that the input state was |ψj〉.Measurements of this type have been carried out by Mohseni, et al. [23].

In the case that the failure space is two-dimensional, it is sometimes pos-sible to obtain some information about the input state even if the initialmeasurement fails [21,22]. Sun, et al. presented an example of an optical net-work that does the following [22]. It consists of an optical 10 port followedby a 6 port. The first three inputs of the 10 port are where the state |ψj〉,j = 1, 2, 3 is sent in, and the other two are in the vacuum state The fail-ure space for this particular situation is two-dimensional, and if the photonemerges from outputs 4 or 5, the measurement has failed. If it emerges from


outputs 1, 2, or 3, we know what the input state was. The 6 port has as itsthree inputs the two failure outputs from the 10 port, and the vacuum. It isconstructed so that if the photon comes out of the first output, we know thatthe input state was not |ψ3〉, if it comes out of the second output, the inputwas not |ψ2〉, and if it comes out of the third output, the measurement hasfailed. Therefore, even if the initial measurement (the 10 port) fails, there issome possibility of gaining information about the input state by further pro-cessing. Note that this depends on the failure space having two dimensions;with a one-dimensional failure space, no further processing is possible.

There is another case where the exact solution to the unambiguous dis-crimination problem is known, and to which we now turn our attention. Firstwe note that a set of N states is called symmetric [3,5] if there exists a unitaryoperator, V , such that, for j = 1, . . . N − 1,

V |ψj〉 = |ψj+1〉 , V |ψN 〉 = |ψ1〉 . (11.34)

This implies that |ψj〉 = V j−1|ψ1〉. The case of unambiguous state discrim-ination for N symmetric states was analyzed by Chefles and Barnett [24].They found an analytical expression for the optimal success probabilities forthe case when the a priori probabilities of the states are the same. The vectors|ψj〉, j = 1, . . . N are now assumed to be linearly independent, and to spanthe entire space. Because the states |ψj〉 form a basis for the space, (11.34)now implies that V N = I. This, in turn, means that V can be expressed as

V =N−1∑k=0

e2πik/N |γk〉〈γk| , (11.35)

where |γk〉 is the eigenstate of V with eigenvalue e2πik/N . The states we aretrying to distinguish among can now be expanded as

|ψj〉 =N−1∑k=0

e2πik(j−1)/Nck|γk〉 . (11.36)

The optimal success probability was found to be

P = N min |ck|2 , (11.37)

where the minimum is taken over k, and throughout the derivation it wasassumed that the a priori probabilities of the states were the same.

11.2.5 The Discrimination of Mixed States

We begin this part by introducing some terminology first. This will greatlyfacilitate the presentation of the material to follow and help us to interpretsome of the existing statements about this topics in the literature.


The support of a mixed state, described by a density matrix, is the spacespanned by its eigenvectors with nonzero eigenvalues. The rank of a mixedstate is the dimension of its support. The kernel of a mixed state is thespace orthogonal to its support. With these definitions at hand, we are readyto interpret the statement that “... one cannot unambiguously discriminatemixed states (the reason is that the IDP scheme does not work for linearlydependent states)” [25]. This is not the only statement of this sort but isa representative one that is obviously correct. But we have to elaborate itfurther. It clearly refers to mixed states that have the same support andexcludes mixed states that have a nonzero overlap with the kernels of theothers. In the following we will focus our attention on precisely those cases,admittedly a small subset of all mixed states of a system, where the densityoperators to be discriminated have different supports. It is only within thissubset that unambiguous discrimination of mixed states is possible.

By the time of the publication of [25], several other works have beenpublished that can be interpreted as special instances of unambiguous dis-crimination of mixed states. Below we give a brief overview of recent progressin this area.

Unambiguous Filtering

In [26], we introduced the problem of unambiguous discrimination betweensets of states. The set of N given states is divided into subsets, and we wantto determine to which subset a particular input state, known to be preparedin one of the N given states, belongs. In the simplest case, the division isinto two sets only, the first containing the first M states and the second theremaining N −M states. We want to unambiguously assign a given inputstate to one of these two subsets. Clearly, the discrimination of two purestates corresponds to N = 2 and M = 1, there is one state in each set. Thenext simplest case is the one with N = 3 and M = 1. This is the case ofunambiguously discriminating whether a state is |ψ1〉 or whether it is in theset {|ψ2〉, |ψ3〉} with a priori probabilities η1, η2, and η3, respectively. Sincein this case all we are interested in is whether a particular input is |ψ1〉 ornot we termed this case unambiguous quantum state filtering. (Actually, weintroduced the term quantum state filtering first in the context of minimum-error discrimination [63] and this work will be discussed in Sect. 11.3.4.) First,it is straightforward to see that filtering is a particular instance of mixed statediscrimination and, by a simple extension of the following considerations,set discrimination, in general, is equivalent to the discrimination of mixedstates. Indeed, since we do not want to resolve the states within a set, we canintroduce the density operators

ρα = |ψ1〉〈ψ1|, ρβ =η2

η2 + η3|ψ2〉〈ψ2|+

η3η2 + η3

|ψ3〉〈ψ3|, (11.38)


with a priori probabilities ηα = η1, ηβ = η2 + η3. Filtering is, thus, thediscrimination between these two density operators, one a rank one mixedstate which is, in fact, a pure state, and the other a rank two mixed state.

Key to the solution is the observation that the ancilla space is one dimen-sional in the Neumark implementation of the optimal POVM. This immedi-ately yields the condition analogous to (11.7),

q1qi = |〈ψ1|ψi〉|2 , (11.39)

where q1 and qi (i = 2, 3) are the failure probabilities for the correspondinginput states. The derivation of the optimum average failure probability isthen very similar to the derivation of the IDP result for two pure states. Theoptimal failure probability, Qopt, is given as

Qopt =

QPOVM if F 2

1+F 2 ≤ ηα ≤ 11+F 2 ,

Qα if ηα < F 2

1+F 2 ,

Qβ if 11+F 2 < ηα .

(11.40)

Here F =√〈ψ1|ρβ |ψ1〉 is the fidelity between a pure and a mixed state [1],

and

QPOVM = 2√ηαηβF ,

Qα = ηα + ηβF 2 ,

Qβ = ηα‖ψ‖1‖2 + ηβ

F 2

‖ψ‖1‖2

. (11.41)

In the last line |ψ‖1〉 is the component of |ψ1〉 in the support of ρβ . Written

in terms of the fidelity between a pure and a mixed state, the solution re-mains valid for an arbitrary number of states in the second set and, so, itrepresents the solution for a rank one vs. rank N unambiguous discriminationproblem where N is arbitrary [27]. An interesting application of this resultfor a probabilistic quantum algorithm will be presented in Sect. 11.2.6. Wenote that unambiguous discrimination between multiple sets of pure statesis a straightforward generalization and was independently investigated byZhang and Ying in [28]. It, too, can be recast in a form of discriminatingbetween mixed states. The problem of filtering a mixed state out of manywas addressed by Takeoka, et al. in [29].

State Comparison

In [30], Barnett, Chefles, and Jex introduced the problem of state comparisonthat can be stated as follows. Given two systems, each of which is in oneof two, in general, nonorthogonal states {|ψ1〉, |ψ2〉}, what is the optimumprobability to determine whether the two systems were prepared in the same


state or in different states? One of the surprising aspects of their result is thatone can decide with a certain probability whether the two systems are not inthe same state even if there is no prior knowledge of the possible states. Inthis context we note that discrimination between non perfectly known stateshas also been addressed by Jezek in [31].

More importantly for our purpose, the comparison of states can be castto a form that corresponds to the discrimination between mixed states de-scribed by rank two density matrices. Obviously, we now want to discriminatebetween the sets {|ψ1〉|ψ1〉, |ψ2〉|ψ2〉} and {|ψ1〉|ψ2〉, |ψ2〉|ψ1〉}. The first setcontains the combined states where the two systems are in the same indi-vidual state with a priori probabilities η1 and η2 for the respective combinedstates and the second set contains the states where the two systems are indifferent individual states with a priori probabilities η3 and η4 for the respec-tive combined states. Introducing density operators for the sets, by employinga slight extension of the method in (11.38), we see that state comparison isequivalent to the discrimination of two rank two mixed states [32]. In [33], wederived the optimal POVM failure probability solution for the unambiguousdiscrimination between a rank two and a rank N density matrix,

QPOVM = 2√ηαηβF , (11.42)

where F is the fidelity between the two mixed states ρα and ρβ [1]. In thecase of state comparison, the intersection between the two supports is onedimensional and the fidelity can easily be calculated to give F = |〈ψ1|ψ2〉|.Thus, for the optimal failure probability of state comparison, QSC , we get

QSC = 2√ηαηβ |〈ψ1|ψ2〉| , (11.43)

where ηα = η1 + η2 and ηβ = η3 + η4 are the a priori probabilities for therespective mixed states. This result slightly generalizes the result of [30],based on a previously unpublished result of our own [33]. In a subsequentwork [34], Jex, Andersson, and Chefles extended the state comparison resultsfrom two systems to the comparison of many. By employing similar methodsto the ones here, this can be shown to be equivalent to the discriminationbetween density matrices of higher rank.

Two Arbitrary Mixed States: General Considerations

Generalizing the ideas of [26]- [34], two important works have appeared re-cently. In the first, Rudolph, et al. [35] established lower and upper boundson the minimum failure probability for the unambiguous discrimination oftwo mixed states. Based on Uhlmann’s theorem [1] they found that the lowerbound is given by the fidelity, and the upper bound is based on the geometri-cal invariants between the kernels. They showed that for all known solutionsthe upper and lower bounds coincide and found numerically in other casesthat the two bounds are very close. As an application of their method, they


also provided the general solution for the one-dimensional kernel problemand applied it to the special case of two rank N − 1 density matrices in anN -dimensional Hilbert space.

Raynal, et al. [36], introduced reduction theorems for the problem of op-timal unambiguous discrimination of two general density matrices of rank Nand M . In particular, they showed that the problem can be reduced to thediscrimination of two density matrices that have the same rank N0 whereN0 is bounded by N0 ≤ min(N,M). Necessary and sufficient conditions foroptimality were discussed also in [25] and [37], along with some numericalmethods based on linear programming.

The upper and lower bounds on the failure probabilities by [35] and thereduction theorems by [36] bring us very close to a full solution of the unam-biguous discrimination problem between two arbitrary mixed states and it isfully expected that a full solution will be found in the near future.

11.2.6 Selected Applications

We shall conclude this section with a discussion of two applications of unam-biguous state discrimination. The first is a quantum cryptographic protocol,while the second is a quantum algorithm.

Bennett proposed using the unambiguous discrimination of two nonorthog-onal states as the basis of a quantum-key-distribution protocol [38]. Alice andBob want to establish a secure key that they can use to send encrypted mes-sages to each other. To do so, Alice sends Bob a sequence of particles, whereeach particle is either in the state |ψ1〉 or |ψ2〉. The state |ψ1〉 correspondsto a bit value of 0 and |ψ2〉 corresponds to a bit value of 1. These states areknown to both Alice and Bob, and they are not orthogonal. Upon receiving aparticle, Bob applies the optimal two-state unambiguous measurement proce-dure to it. He then tells Alice over a public channel whether the measurementsucceeded or failed. If it succeeded, they keep the bit, and if it failed, theydiscard the bit. In this way they can establish a key.

An eavesdropper, Eve, who wants to determine the key without beingdiscovered has a problem. Let us assume that she can intercept the particlesAlice is sending to Bob, and that she knows the states |ψ1〉 and |ψ2〉. Becausethese states are not orthogonal, she cannot tell with certainty which state aparticular particle is in. One possibility is for her to apply the same procedureused by Bob, optimal two-state unambiguous state discrimination. If hermeasurement succeeds, all is well. She simply notes which state she found,and prepares another particle in this state and sends it on to Bob. She thenknows this key bit, and Alice and Bob do not know that she knows. However,if her measurement fails, she does not know which state Alice sent, and shehas to make a guess which state to send on to Bob. That means that she willintroduce discrepancies between the state that Alice sent and the state thatBob received. In some of the cases in which this happens, Bob’s measurementwill succeed, and this will cause errors in the key to appear. If Alice and Bob


publicly share a subset of their good key bits (these bits have to then bediscarded), and if they see discrepancies, then they know an eavesdropperwas present, and that the key is insecure. If they find none, then with highprobability (Eve could get lucky and have all of her measurements succeed,but this is very improbable) the key is secure.

Our second example is a probabilistic quantum algorithm to discriminatebetween sets of Boolean functions [27]. A Boolean function on n bits is onethat returns either 0 or 1 as output for every possible value of the inputx, where 0 ≤ x ≤ 2n − 1. The function is constant if it returns the sameoutput on all of its arguments, i.e. either all 0’s or all 1’s; it is balancedif it returns 0’s on half of its arguments and 1’s on the other half; and itis biased if it returns 0’s on m0 of its arguments and 1’s on the remainingm1 = 2n−m0 arguments (m0 �= m1 �= 0 or 2n−1). Classically, if one is givenan unknown function and told that it is either balanced or constant, one needs2(n−1)+1 measurements to decide which. Deutsch and Jozsa [39] developed aquantum algorithm that can accomplish this task in one step. To discriminatea biased Boolean function from an unknown balanced one, 2(n−1) +m1 + 1measurements are needed classically, where, without loss of generality, wehave assumed that m1 < m0. There is a probabilistic quantum algorithm,based on quantum state filtering, that can unambiguously discriminate aknown biased Boolean function from a given set of balanced ones. There is asignificant chance that only one function evaluation will be necessary.

The algorithm distinguishes between sets of Boolean functions. Let f(x),where 0 ≤ x ≤ 2n−1, be a Boolean function, i.e. f(x) is either 0 or 1. One ofthe sets we want to consider is a set of balanced functions. The second set hasonly two members, and we shall call it Wk. A function is in Wk if f(x) = 0for 0 ≤ x < [(2k − 1)/2k]2n and f(x) = 1 for [(2k − 1)/2k]2n ≤ x ≤ 2n − 1,or if f(x) = 1 for 0 ≤ x < [(2k − 1)/2k]2n and f(x) = 0 for [(2k − 1)/2k]2n ≤x ≤ 2n − 1. We now wish to distinguish between the balanced functions andfunctions in Wk, that is, we are given an unknown function that is in oneof the two sets, and we want to find out which set it is in. We note thatthe two functions in Wk are biased functions, so that this is a special caseof a more general problem of distinguishing a set of biased functions frombalanced functions.

The Deutsch-Jozsa algorithm makes use of the unitary operation

|x〉|y〉 → |x〉|y + f(x)〉, (11.44)

where the first state, |x〉, is an n-qubit state, the second state, |y〉, is a singlequbit state, and the addition is modulo 2. The state |x〉, where x is an n-digit binary number, is a member of the computational basis for n qubits,and the state |y〉, where y is either 0 or 1, is a member of the computationalbasis for a single qubit. In solving the Deutsch-Jozsa problem, this mappingis employed in the following way


D−1∑x=0

|x〉(|0〉 − |1〉) →D−1∑x=0

(−1)f(x)|x〉(|0〉 − |1〉), (11.45)

where D = 2n. This has the effect of mapping Boolean functions to vectorsin the D-dimensional Hilbert space, HD, and we shall do the same. The finalqubit is not entangled with the remaining n qubits and can be discarded.The vectors

∑D−1x=0 (−1)f(x)|x〉 that are produced by balanced functions are

orthogonal to those produced by constant functions. This is why the Deutsch-Jozsa problem is easy to solve quantum mechanically. In our case, the vectorsproduced by functions in Wk are not orthogonal to those produced by bal-anced functions. However, unambiguous quantum state filtering provides aprobabilistic quantum algorithm for the optimal solution of this problem.

In order to apply the filtering solution, we note that both functions inWkare mapped, up to an overall sign, to the same vector in HD, which we shallcall |wk〉. The vectors that correspond to balanced functions are containedin the subspace, Hb, of HD, where Hb = {|v〉 ∈ HD|

∑D−1x=0 vx = 0}, and

vx = 〈x|v〉. This subspace has dimension 2n − 1 = D − 1, and it is possibleto choose an orthonormal basis, {|vi〉|i = 2, . . . D}, for it in which each basiselement corresponds to a particular balanced Boolean function [40].

Let us first see how the filtering procedure performs when applied to theproblem of distinguishing |wk〉(= |ψ1〉) from the set of the D−1 orthonormalbasis states, |vi〉(= |ψi〉), in Hb. We assume their a priori probabilities to beequal, i.e. ηi = η = (1− η1)/(D − 1) for i = 2, . . . D, where η1 is the a prioriprobability for |wk〉. For ‖ψ‖

1‖2 = ‖w‖k‖2 ≡ fk we obtain fk = (2k−1)/22k−2.

Then the average overlap, Sk, between |wk〉 and the set of balanced basisvectors can be written as

Sk =1− η1D − 1

fk , (11.46)

in terms of fk [40]. The failure probabilities are given by the filtering re-sult, using S = Sk and, to good approximation, the POVM result holdswhen 1/2k−2 ≤ Dη1 ≤ 2k−2. For example, in the case in which all of thea priori probabilities are equal, i.e. η1 = 1/D, we find that Q1 = Q2 ≡QSQM = (1 + fk)/D where SQM stands for Standard Quantum Measure-ment (or von Neumann projective measurement). To good approximation,QPOVM/QSQM = 4/2k/2, which, for k � 1, shows that the POVM canperform significantly better than the von Neumann measurements.

Now that we know how this procedure performs on the basis vectorsin Hb, we shall examine its performance on any balanced function, i.e. weapply it to the problem of distinguishing |wk〉 from the set of all states inHb that correspond to balanced functions. The number of such states is N =D!/(D/2)!2 and we again assume their a priori probabilities to be equal,η = (1− η1)/N . It can be shown [40] that the average overlap between |wk〉and the set {|v〉} is given by the same expression, (11.46), as in the previouscase. Therefore, much of what was said in the previous paragraph remains


valid for this case, as well, with one notable difference. The case η1 = 1/D nowdoes not correspond to equal a priori probability for the states but, rather,to a priori weight of the sets that is proportional to their dimensionality. Inthis case it is the POVM that performs best. In the case of equal a prioriprobability for all states, η1 = 1/(N + 1), we are outside of the POVMrange of validity and it is the first standard quantum measurement (SQM1)that performs best. Both the POVM and the SQM1 are good methods fordistinguishing functions in Wk from balanced functions. Which one is betterdepends on the a priori probabilities of the functions.

Classically, in the worst case, one would have to evaluate a function2n[(1/2)+(1/2k)]+1 times to determine if it is in Wk or if it is an even func-tion. Using quantum information processing methods, one has a very goodchance of determining this with only one function evaluation. This showsthat Deutsch-Jozsa-type algorithms need not be limited to constant func-tions; certain kinds of biased functions can be discriminated as well.

11.3 State Discrimination with Minimum Error

11.3.1 Introductory Remarks

In the previous chapter we have required that, whenever a definite answer isreturned after a measurement on the state, the result should be unambigu-ous, at the expense of allowing inconclusive outcomes to occur. For manyapplications in quantum communication, however, one wants to have conclu-sive results only. This means that errors are unavoidable when the states arenon-orthogonal. Based on the outcome of the measurement, in each singlecase then a guess has to be made as to what the state of the quantum systemwas. This procedure is known as quantum hypothesis testing. The problemconsists in finding the optimum measurement strategy that minimizes theprobability of errors.

Let us state the optimization problem a little more precisely. In the mostgeneral case, we want to distinguish, with minimum probability of error, be-tween N given states of a quantum system (N ≥ 2), being characterized bythe density operators ρj (j = 1, 2, . . . , N) and occurring with the given a pri-ori probabilities ηj which sum up to unity. The measurement can be formallydescribed with the help of a set of detection operators Πj that refer to thepossible measurement outcomes [3, 4]. They are defined in such a way thatTr(ρΠj) is the probability to infer the system is in the state ρj if it has beenprepared in a state ρ. Since the probability is a real non-negative number,the detection operators have to be Hermitean and positive-semidefinite. Inthe error-minimizing measurement scheme the measurement is required tobe exhaustive and conclusive in the sense that in each single case one of theN possible states is identified with certainty and inconclusive results do not


occur. This leads to the requirement

N∑j=1

Πj = IDS, (11.47)

where IDSdenotes the unit operator in the DS-dimensional physical state

space of the quantum system. The overall probability Perr to make an erro-neous guess for any of the incoming states is then given by

Perr = 1− Pcorr = 1−N∑j=1

ηjTr(ρjΠj) (11.48)

with∑j ηj = 1. Here we introduced the probability Pcorr that the guess is

correct. In order to find the minimum-error measurement strategy, one hasto determine the specific set of detection operators that minimizes the valueof Perr under the constraint given by (11.47). By inserting these optimum de-tection operators into (11.48), the minimum error probability Pmin

err ≡ PE isdetermined. The explicit solution to the error-minimizing problem is not triv-ial and analytical expressions have been derived only for a few special cases.

11.3.2 Distinguishing Two Quantum States with Minimum Error

The Helstrom Formula

For the case that only two states are given, either pure or mixed, the mini-mum error probability, PE , was derived in the mid 70s by Helstrom [3] in theframework of quantum detection and estimation theory. We find it instruc-tive to start by analyzing the two-state minimum-error measurement withthe help of an alternative method (cf. [41, 42]) that allows us to gain imme-diate insight into the structure of the optimum detection operators, withoutapplying variational techniques. Starting from (11.48) and making use of therelations η1 + η2 = 1 and Π1 +Π2 = IDS

that have to be fulfilled by the apriori probabilities and the detection operators, respectively, we see that thetotal probability to get an erroneous result in the measurement is given by

Perr = 1−2∑j=1

ηjTr(ρjΠj) = η1Tr(ρ1Π2) + η2Tr(ρ2Π1). (11.49)

This can be alternatively expressed as

Perr = η1 + Tr(ΛΠ1) = η2 − Tr(ΛΠ2), (11.50)

where we introduced the Hermitean operator

Λ = η2ρ2 − η1ρ1 =DS∑k=1

λk|φk〉〈φk|. (11.51)


Here the states |φk〉 denote the orthonormal eigenstates belonging to theeigenvalues λk of the operator Λ. The eigenvalues are real, and without lossof generality we can number them in such a way that

λk < 0 for 1 ≤ k < k0,λk > 0 for k0 ≤ k ≤ D,λk = 0 for D < k ≤ DS . (11.52)

By using the spectral decomposition of Λ, we get the representations

Perr = η1 +DS∑k=1

λk〈φk|Π1|φk〉 = η2 −DS∑k=1

λk〈φk|Π2|φk〉. (11.53)

Our optimization task now consists in determining the specific operators Π1,or Π2, respectively, that minimize the right-hand side of (11.53) under theconstraint that

0 ≤ 〈φk|Πj |φk〉 ≤ 1 (j = 1, 2) (11.54)

for all eigenstates |φk〉. The latter requirement is due to the fact that Tr(ρΠj)denotes a probability for any ρ. From this constraint and from (11.53) itimmediately follows that the smallest possible error probability, Pmin

err ≡ PE ,is achieved when the detection operators are chosen in such a way that theequations 〈φk|Π1|φk〉 = 1 and 〈φk|Π2|φk〉 = 0 are fulfilled for eigenstatesbelonging to negative eigenvalues, while eigenstates corresponding to positiveeigenvalues obey the equations 〈φk|Π1|φk〉 = 0 and 〈φk|Π2|φk〉 = 1. Hencethe optimum detection operators can be written as

Π1 =k0−1∑k=1

|φk〉〈φk|, Π2 =DS∑k=k0

|φk〉〈φk|, (11.55)

where the expression for Π2 has been supplemented by projection opera-tors onto eigenstates belonging to the eigenvalue λk = 0, in such a waythat Π1 + Π2 = IDS

. Obviously, provided that there are positive as wellas negative eigenvalues in the spectral decomposition of Λ, the minimum-error measurement for discriminating two quantum states is a von Neumannmeasurement that consists in performing projections onto the two orthog-onal subspaces spanned by the set of states {|φ1〉, . . . , |φk0−1〉}, on the onehand, and {|φk0〉, . . . , |φDS

〉}, on the other hand. An interesting special casearises when negative eigenvalues do not exist. In this case it follows thatΠ1 = 0 and Π2 = IDS

which means that the minimum error probabilitycan be achieved by always guessing the quantum system to be in the stateρ2, without performing any measurement at all. Similar considerations holdtrue in the absence of positive eigenvalues. We note that these findings are in


agreement with the recently gained insight [44] that measurement does not al-ways aid minimum-error discrimination. By inserting the optimum detectionoperators into (11.50) the minimum error probability is found to be [42]

PE = η1 −k0−1∑k=1

|λk| = η2 −D∑k=k0

|λk|. (11.56)

Taking the sum of these two alternative representations and using η1+η2 = 1,we arrive at

PE =12

(1−

∑k

|λk|)

=12

(1− Tr|Λ|) , (11.57)

where |Λ| =√Λ†Λ. Together with (11.48) this immediately yields the well-

known Helstrom formula [3] for the minimum error probability in discrimi-nating ρ1 and ρ2,

PE =12

(1− Tr|η2ρ2 − η1ρ1|) =12

(1− ‖η2ρ2 − η1ρ1‖) . (11.58)

In the special case that the states to be distinguished are the pure states|ψ1〉 and |ψ2〉, this expression reduces to [3]

PE =12

(1−

√1− 4η1η2|〈ψ1|ψ2〉|2

). (11.59)

The set-up of the detectors that achieve the optimum error probabilities isparticularly simple for the case of equal a priori probabilities. Two orthogo-nal detectors, placed symmetrically around the two pure states, will do thetask, as shown in Fig. 11.6. The simplicity is particularly striking when onecompares this set-up to that of Fig. 11.3, that displays the correspondingPOVM set-up for optimal unambiguous discrimination.

In order to provide a simple and intuitive physical picture of an error-minimizing state discrimination measurement, let us consider the experi-ment performed by Barnett and Rijs [43] for distinguishing two equiprobablenon-orthogonal single-photon polarization states, produced with the help ofstrongly attenuated light pulses. Imagine, we are given a series of very weaklight pulses and we know in advance that, by preparation, each of the pulsesis linearly polarized, with equal probability either in the direction e1 or inthe direction e2 with e1,2 = cos θ ex ± sin θ ey, where 0 ≤ θ ≤ π/4. Bothkinds of pulses are assumed to have equal intensity and to contain on theaverage much less than one photon, so that the probability to obtain twophotodetector clicks from a single pulse is negligible. Using a linear polarizer,we want to determine for each photon which polarization it had, by mak-ing the guess that it had polarization e1 when the photon is transmitted bythe polarizer and had polarization e2 when the photon is absorbed in the


Fig. 11.6. Detector configuration for the optimum minimum-error discriminationof two pure states with equal a priori probabilities. A von Neumann measure-ment with two orthogonal detectors placed symmetrically around |ψ1〉 and |ψ2〉will achieve the optimum.

polarizer. The problem of finding the optimum measurement strategy thenamounts to determining the optimum orientation of the polarizer that min-imizes the probability to make a wrong guess. It is easy to calculate thatthe error probability is smallest when the polarizer is oriented symmetricallywith respect to the two given polarization directions, i. e. when its transmis-sion direction is given by the unit vector ep = (ex + ey)/

√2. In this case

the fraction of the intensity that is transmitted from the first pulse, whichis also the probability, that the photon is indeed transmitted provided it isof the first kind, follows from the projection of e1 onto ep and is given bycos2(π/4 − θ). The same quantity describes the probability that the photonis absorbed, provided it is of the second kind, as can be found by projectinge2 onto the vector perpendicular to ep. Hence the resulting error probabilityis determined by

PE = 1− cos2(π

4− θ) =

12(1− sin 2θ) =

12(1−

√1− |e1e2|2). (11.60)

which reproduces (11.59) with η1 = η2 = 1/2 when the scalar product e1e2is replaced by the overlap 〈ψ1|ψ2〉.

Minimum-Error DiscriminationVersus Unambiguous Discrimination for Two Mixed States

In the error-minimizing scheme for discriminating two different mixed statesρ1 and ρ2 of a quantum system, a non-zero probability of making a correct


guess can always be achieved. However, it is clear that we can only distinguishthe two mixed states unambiguously when, in the DS-dimensional physicalstate space of the quantum system, there exists at least one state vector thathas a non-zero probability of occurring in the first of the mixed states but willnever be found in the second mixed state. Recently it has been shown thatthe maximum success probability for unambiguously discriminating the twogiven mixed states does not exceed a certain upper bound [35], depending onthe states and their a priori probabilities. Consequently, the minimum failureprobability, QF , cannot be smaller than a certain lower bound, QL. We nowinvestigate the relation between this lower bound on the failure probability,on the one hand, and the minimum error probability for discriminating thesame two states, on the other hand. The procedure that we shall apply isclosely related to the derivation of inequalities between the fidelity and thetrace distance [1].

Let us first consider the failure probability QF for unambiguously discrim-inating the two mixed states. From the result derived in [35] it follows that

QF ≥{

2√η1η2 F (ρ1, ρ2) ≡ QL if F (ρ1, ρ2) <

√ηminηmax

ηmin + ηmax[F (ρ1, ρ2)]2 ≥ QL otherwise,(11.61)

where in the second line we have used the fact that the arithmetic meancannot be smaller than the geometric mean. Here ηmin(ηmax) is the smaller(larger) of the two a priori probabilities η1 and η2, and F is the fidelity definedas F (ρ1, ρ2) = TrO, where O =

(√ρ2 ρ1

√ρ2)1/2. To allow for a comparison

between PE and QF , or PE and QL, respectively, we need a suitable orthonor-mal basis. It has been shown [1,41] that when the basis states are chosen to bethe eigenstates {|l〉} of the operator ρ−1/2

2 Oρ−1/22 , the fidelity takes the form

F (ρ1, ρ2) =∑l

√〈l|ρ1|l〉〈l|ρ2|l〉 =

∑l

√rl sl. (11.62)

Here∑l|l〉〈l| = I, and we introduced the abbreviations rl = 〈l|ρ1|l〉 and sl =

〈l|ρ2|l〉. The lower bound on the failure probability then obeys the equation

1−QL = 1− 2√η1η2

∑l

√rl sl =

∑l

(√η1rl −

√η2sl)

2, (11.63)

where the second equality sign is due to the relation η1 + η2 = 1 and to thenormalization conditions Trρ1 =

∑l rl = 1 and Trρ2 =

∑l sl = 1.

Now we estimate the minimum error probability PE , using the sameset of basis states {|l〉}. From (11.57) and from the fact that 〈φk|φk〉 =∑l |〈φk|l〉|2 = 1 it follows that

1− 2PE =∑k

|λk| =∑l

∑k

|λk||〈φk|l〉|2

≥∑l

∣∣∣∣∣∑k

λk|〈φk|l〉|2∣∣∣∣∣ =

∑l

|〈l|Λ|l〉|, (11.64)


where the last equality sign is due to the spectral decomposition (11.51).After expressing Λ in terms of the density operators describing the givenstates, we arrive at

1− 2PE ≥∑l

|〈l|η1ρ1 − η2ρ2|l〉|

=∑l

|√η1rl −√η2sl| |

√η1rl +

√η2sl| . (11.65)

By comparing the expressions on the right-hand sides of (11.63) and (11.65),respectively, it becomes immediately obvious that 1− 2PE ≥ 1−QL, or

PE ≤12QL . (11.66)

This means that for two arbitrary mixed states, occurring with arbitrary apriori probabilities, the smallest possible failure probability in unambiguousdiscrimination is at least twice as large as the smallest probability of errorsin minimum-error discrimination of the same states.

11.3.3 The General Strategyfor Minimum-Error State Discrimination

Formal Solution for N Mixed States

We return now to the general problem of discriminating with minimum errorbetween N given mixed states, where N is an arbitrary number. As has beenoutlined already in the introductory remarks, the problem amounts to findingthe specific optimum detection operators that minimize the expression (11.48)under the constraint (11.47). It has been shown by Holevo [45] and Yuen etal. [46] that the set of detection operators {Πj} determining the optimummeasurement strategy must satisfy the necessary and sufficient conditions

Πk(ηkρk − ηjρj)Πj = 0 (11.67)∑j

ηjρjΠj − ηkρk ≥ 0, (11.68)

(1 ≤ j, k ≤ N), where the last equation expresses the fact that the eigenval-ues of the operator on the left-hand side are non-negative. These conditionscan be understood from the following considerations. In order to take theconstraint (11.47) into account, it is possible to introduce a Lagrange opera-tor Γ , in analogy to a Lagrangian multiplier. The optimization task is thenequivalent to maximizing the operator functional

Pcorr({Πj}, Γ ) =∑j

Tr[(ηjρj − Γ )Πj ] + TrΓ, (11.69)


with Γ and each of the Πj varying independently. Since Pcorr is real, Γhas to be Hermitean. The necessary condition for an extremum, δPcorr = 0,yields for any j = 1, . . . , N the requirement that Tr[(ηjρj −Γ )δΠj ] = 0. Thedetection operators can be varied by starting from their eigenstate expansionΠj =

∑l πjl|µjl〉〈µjl| and performing variations in the state vectors, yielding

δΠj =∑l πjl|µjl〉〈δµjl| + H.A., where πjl ≥ 0. This leads to the necessary

extremal condition [25]

(ηjρj − Γ )Πj = 0, (11.70)

from which it follows that for any j, k the equations ηjρjΠj = ΓΠj andηkΠkρk = ΠkΓ

† have to be fulfilled, where Γ is Hermitean. By multiplyingthe first equation from the left withΠk and the second from the right withΠjand taking the difference, the representation (11.67) for the necessary con-dition becomes immediately obvious. Now we discuss the second condition.First we note that

Γ =∑j

ηjρjΠj , (11.71)

which follows from (11.70) and from the constraint∑j Πj = I. As can be seen

from (11.69) and (11.70), the extremal value is given by Pcorr = TrΓ . Cer-tainly for the extremum of Pcorr to be a maximum, its value cannot be smallerthan the probability of correct guesses that would follow from other choices ofthe set of detection operators, in particular from the choice {Πj} = {I δj,k},leading to Pcorr = Tr(ηkρk). This means that for the extremum to be a maxi-mum the relation TrΓ ≥ Tr(ηkρk) has to be fulfilled for any k, which is indeedguaranteed by the requirement (11.68). The conditions (11.67) and (11.68)implicitly determine the solution of the general minimum-error discriminationproblem.

Survey of Explicitly Solvable Special Cases

In the following we give a brief overview that summarizes the cases where ex-plicit analytical expressions for the optimum detection operators, and hencefor the minimum error probability, have been determined from the implicitgeneral solution. The case that only two states are given, N = 2, has beenalready extensively discussed in the previous section. When the two givenstates are pure and occur with equal prior probability, the Helstrom boundderived there is a special case of a more general result, referring to the classof equiprobable and symmetric pure states. For these states the solution ofthe minimum-error discrimination problem, as derived by Ban et al. [47], willbe discussed in a separate section. Recently Eldar et al. [48] and Chou andHsu [49] obtained an extension of this solution to the case of N equiprobablestates that are symmetric and mixed. A few other cases have been solved ana-lytically, too. They include certain classes of linearly independent states [50]


and also equiprobable states the projectors of which sum up to the iden-tity [46]. Moreover, Barnett [51] found the minimum-error strategy for mul-tiple symmetric pure states, and Andersson et al. [52] solved the case ofthree mirror symmetric pure states. With respect to the general problem ofdiscriminating an arbitrary number of mixed states, Hunter [44] found thecondition and gave the solution for those cases when the best strategy con-sists in making no measurement at all, but simply to guess always the statewith the highest a priori probability. We encountered an example of such acase when considering the discrimination between two mixed states.

Apart from the analytical solutions, the minimum-error strategy has beenalso investigated numerically. In particular, Jezek et al. [53] proposed analgorithm for finding the optimum measurement by applying the theory ofsemi-definite programming.

Generalized Measurements

When the N given states of the quantum system are linearly independent,the minimum-error strategy for state discrimination is always a von Neumannmeasurement, as has been recently proved by Eldar [54]. This means thatthe detection operators are mutually orthogonal projection operators in theHilbert space of the quantum system, fulfilling the relation ΠjΠk = δjkΠj ,where 1 ≤ j, k ≤ N . We note that the quantum states are called linearlyindependent when the combined set of all the eigenvectors of the density op-erators ρj (j = 1, . . . , N) forms a set of linearly independent state vectors.In this case the error-minimizing discrimination measurement is a projectivemeasurement that can be realized by performing measurements on the orig-inal quantum system alone. Generally, however, this need not always be thecase. In particular, when the number of different measurement outcomes, orthe number of detection operators, respectively, exceeds the dimensionalityof the physical state space of the quantum system, the detection operatorscannot be represented by projection operators in that state space, as becomesimmediately obvious from the constraint

∑j Πj = IDS

. The state discrimi-nation measurement then has to be described as a generalized measurement,based on positive-operator valued measures, and the detection operators Πjare also called POVM-elements. According to Neumark’s theorem [13], anygeneralized measurement can be realized with the help of a unitary transfor-mation and a projective measurement in an extended Hilbert space.

In order to illustrate these general considerations we find it worthwhileto briefly consider a prominent example for a generalized minimum-errormeasurement, which consists in discriminating between the three states of asingle qubit defined as

|ψ1〉 = −12

(|0〉+

√3 |1〉

),


|ψ2〉 = −12

(|0〉 −

√3 |1〉

),

|ψ3〉 = |0〉, (11.72)

where |0〉 and |1〉 are the orthonormal basis states of the qubit. The threegiven states form an overcomplete set of symmetric states that is known as thetrine ensemble [55]. Provided that the occurrence of each of the three states isequally probable, the optimum detection operators for distinguishing amongthem with minimum error, or the optimum POVM-elements, respectively,are given by [55]

Πj = A†jAj =

23|ψj〉〈ψj |, (11.73)

for j = 1, 2, 3. The probability of correctly identifying any of the trine statesin the error-minimizing measurement is 2/3 and the probability of making anerror is 1/3. Since the states |ψj〉 are normalized, it is clear that the opera-tors Πj do not represent projection operators. For a physical implementationof the generalized measurement, means have to be found for extending theoriginally two-dimensional Hilbert space of the system. Then a unitary trans-formation on the extended system of basis states has to be performed in sucha way that a final von Neumann measurement realizes the specific projectivemeasurement that is necessary for applying Neumark’s theorem.

As discussed before, there are two conceptually different ways of achievingan extension of the dimensionality of the Hilbert space. The first amountsto defining the original quantum system in such a way that auxiliary quan-tum states can be directly added. The extended Hilbert space is then thedirect sum of the Hilbert space spanned by the states of the original systemand of the Hilbert space spanned by the auxiliary states, being also calledancilla states. For the trine ensemble, it is possible to associate the threetwo-dimensional non-normalized detection states Aj =

√2/3 |ψj〉 with three

orthonormal states in three dimensions, given by

|ψj〉 =

√23|ψj〉+

√13|2〉, (11.74)

where the auxiliary normalized state |2〉 is chosen to be orthogonal to thetwo basis states |0〉 and |1〉. By performing the von-Neumann measurementthat consists of the three projections |ψj〉〈ψj | (j = 1, 2, 3) in the enlarged,i. e. three-dimensional Hilbert space, the required generalized measurementis realized in the original two-dimensional Hilbert space of the qubit [56].To implement this scheme, an original atomic qubit system can be definedto consist of only two electronic states of a multi-level atom. The unitarytransformation necessary for the generalized measurement can then be ac-complished with the help of a third level, by appropriately redistributing thepopulation of the three atomic levels using sequences of Raman transitionsinduced by specially taylored classical pulses [56]. Finally the resulting level


population is detected for performing the projective measurement. Explicittheoretical proposals have been made for using this scheme in order to realizeboth optimum unambiguous discrimination between two nonorthogonal purequbit states, and minimum-error discrimination for the trine ensemble [56].Similarly, a single-photon qubit system could be represented by only twoout of three possible input modes, or input ports, respectively, of an opticalnetwork built of beam splitters and phase shifters. The unitary transforma-tion then would be implemented by this linear network, and the projectivemeasurement could be performed by detecting the photon at one of the out-put ports. It is obvious that the direct-sum representation of extending theHilbert space relies on assuming that the original qubit secretly consists oftwo components of a qutrit [57].

The second way of enlarging the Hilbert space relies on coupling the orig-inal system to an auxiliary system, or ancilla system, with the help of aphysical interaction. The Hilbert space of the combined system is the tensorproduct of the Hilbert spaces of both subsystems. As has been stated byJozsa, et al. [58], from a physical point of view the adjoining of an additionalancilla system is the only available means of extending a space while retain-ing the original system intact. The formalism of (nonorthogonal) POVMs isthen a mathematical artifice that expresses the residual effect on the originalsystem when a von Neumann measurement is performed on the combinedsystem after the interaction [58]. Clarke, et al. demonstrated experimentallythe minimum-error discrimination for the trine ensemble represented by po-larization states of a single photon, using an interferometric set-up [55]. Theyalso performed the optimal error minimizing measurements on the tetrad en-semble, which, as its name implies, consists of four states [55]. The pointscorresponding to these states on the Bloch sphere lie on the corners of atetrahedron.

Error-Minimizing Discrimination with a Fixed Numberof Inconclusive Results

As an extension of the measurement strategy described so far, the errorminimizing discrimination strategy has been also studied under the condi-tion that inconclusive results are now allowed to occur, but with a fixedprescribed probability. This probability is assumed to be smaller than theminimum failure probability resulting from the measurement scheme for op-timum unambiguous discrimination, where errors do not occur at all. Theoptimum measurement minimizing the error under this prescribed conditionis then intermediate between generic minimum-error discrimination and op-timum unambiguous discrimination. It was first investigated for pure statesby Chefles and Barnett [59] and by Zhang et al. [60]. Later the method wasgeneralized to the case of mixed states by Fiurasek and Jezek [25] and by El-dar [37]. The possible occurrence of inconclusive results has to be accountedfor in the basic equations by introducing an additional detection operator Π0


such that Tr(ρΠ0) describes the probability to get an inconclusive outcomeprovided the system is in a state ρ. The given fixed value of this probabil-ity, or the given failure probability, respectively, modifies the constraint onthe detection operators. By optimization, the minimum probability for theoccurrence of errors in the conclusive outcomes can be determined.

11.3.4 Selected Problems of Minimum-Error Discrimination

Distinguishing N Symmetric Pure States

We conclude our review of minimum-error discrimination by a more detailedconsideration of a few special problems. In particular, in this connection wealso present the results of some of our own recent investigations. To beginwith, we recall a pure-state discrimination problem that is exactly solvableand has found wide application in quantum communication. It consists in theso called square-root measurement that discriminates with minimum errorbetween N equally probable symmetric states. Symmetric pure states aredefined in such a way that each state results from its predecessor by applyinga unitary operator V in a cyclic way [47],

|ψj〉 = V |ψj−1〉 = V j−1|ψ1〉, |ψ1〉 = V |ψN 〉, (11.75)

implying that V N = I. For the case that the states occur with equal a prioriprobability, i. e. that ηj = 1/N for each of the states, Ban et al. found thatthe optimum detection operators for minimum-error discrimination are givenby [47]

Πj = A†jAj = B−1/2|ψj〉〈ψj |B−1/2 ≡ |µj〉〈µj |, (11.76)

where

B =N∑j=1

|ψj〉〈ψj |. (11.77)

The states |µj〉 = B−1/2|ψj〉 are in general non-normalized and are calleddetection states. It is obvious that the special structure of the detectionoperators, or of the detection states, respectively, suggests the name “square-root-measurement”. The minimum error probability PE for this measurementis [47]

PE = 1− 1N

N∑j=1

|〈µj |ψj〉|2, (11.78)

in accordance with the fact that in the corresponding optimized measurementscheme the quantum system is inferred to have been prepared in the state


|ψj〉 provided that the state |µj〉 is detected. When the detection states |µj〉are orthonormal, the detection operators are projection operators and theminimum-error measurement is a von-Neumann measurement, otherwise itis a generalized measurement. The latter always holds true when the numberof states exceeds the dimensionality of the physical state space of the quantumsystem, as can be immediately seen from the fact that the detection operatorshave to sum up to the unit operator in that space. In this case the given statesare linearly dependent and form an overcomplete set in the Hilbert space ofthe system.

Let us apply the general solution in order to investigate minimum-errordiscrimination for the set of the N symmetric states

|ψj〉 =D∑k=1

ck ei 2πN j(k−1) |γk〉, (N ≥ D), (11.79)

where the coefficients ck are arbitrary non-zero complex numbers with∑k |ck|2 = 1, and the states |γk〉 (k = 1, . . . , D) form a D-dimensional

orthonormal basis. The given symmetric states are non-orthogonal exceptfor the case that both the conditions N = D and |ck|2 = 1/N are fulfilled.For distinguishing them with minimum error, provided that they occur withequal a priori probability, we obtain the optimum detection states

|µj〉 =1√N

D∑k=1

ck|ck|

ei 2πN jk |γk〉, (11.80)

yielding the minimum error probability [61]

PE = 1− 1N

(D∑k=1

|ck|)2

. (11.81)

When N > D the given symmetric states are linearly dependent and forman overcomplete set. In this case the detection states |µj〉 are non-orthogonaland non-normalized, with 〈µj |µj〉 = D/N . When N = D, however, the states|ψj〉 are linearly independent and therefore can be discriminated also unam-biguously, as has been pointed out in the corresponding section of this review.Assuming equal a priori probabilities, the minimum failure probability, QF ,for unambiguous discrimination of symmetric states has been derived to beQF = 1 − N min|ck|2 [24]. Comparing this with the expression for PE , theminimum error probability is found to be smaller than QF . It is worth men-tioning that the minimum error probability, PE , on the one hand, and thefailure probability, QF , on the other hand, have been considered as distin-guishability measures for ordering different ensembles of N equally probablesymmetric pure states, and it has been found that these two measures imposedifferent orderings [62].


We used the preceding general results for studying minimum-error dis-crimination betweenm-photon polarization states, referring to a fixed numberm of indistinguishable photons (m = 1, 2, . . . ). These states are superpositionstates of the m+ 1 orthonormal polarization basis states

|γ(m)k 〉 =

1√(m− k)!k!

(a†1)m−k(a†

2)k|0〉 ≡ |m− k, k〉, (11.82)

where k = 0, 1, . . .m. The basis states correspond to the m+ 1 different pos-sibilities of distributing m indistinguishable photons among two orthogonalpolarization modes, characterized by the photon creation operators a†

1 anda†2, respectively, and |0〉 is the vacuum state of the field. A specific set of

symmetric m-photon polarization states is defined by

|ψ(m)j 〉 =

1√m!

(cos θ a†

1 + sin θ ei 2πN j a†

2

)m|0〉, (11.83)

where j = 1, . . . N . Interestingly, for N = 4 and θ = π/4 these symmetricstates are identical with the states that result when the standard protocol forquantum key distribution [65] is applied to m-photon pulses. Assuming againequal a priori probabilities, the minimum-error probability for discriminatingthe states reads for m ≤ N − 1

P(m)E (θ) = 1− 1

N

[m∑k=0

√(m

k

)cosm−k θ sink θ

]2

. (11.84)

For the case of two-photon-polarization states, m = 2, we found [61] that forθ = π/4 the minimum error probability is given by PE = 1− (3+2

√2)/(2N)

which is smaller than the value that would result for the corresponding single-photon polarization states, being PE = 1−2/N . In general, the states |ψ(m)

j 〉can be considered to consist of m identical copies of indistinguishable qubits,or photons, respectively, being each in the state cos θ|1, 0〉 + sin θei 2π

N j |0, 1〉.Therefore our results show that by performing a joint measurement on mcopies, instead of a measurement on a single copy only, the probability ofmaking a correct guess for the actual state can be enhanced. However, inorder to physically realize a joint measurement of this kind, in many casesan m-photon interaction process would be necessary, while a measurementon a single copy can always be achieved much more simply with the helpof linear optics. We also gave a recipe for a linear optical multiport per-forming the generalized measurement that discriminates with minimum er-ror between N equiprobable symmetric single-photon polarization states andwe discussed how the corresponding two-photon-polarization states could bediscriminated, at least in principle, with the help of polarization-dependenttwo-photon absorption [61].


Subset Discrimination and Quantum Filteringin a Two-Dimensional Hilbert Space

While in the previous section we dealt with distinguishing between N individ-ual pure states, we now turn to the error-minimizing discrimination betweentwo subsets of a given set of N pure states. In our work [63] we studiedthis task for two sets of linearly dependent pure states that collectively spanonly a two-dimensional Hilbert space. Let us formulate the problem moreprecisely. We want to devise a measurement that allows us to decide, withthe smallest possible error and without inconclusive answers, whether theactual state of the system belongs to the subset of states {|ψ1〉, . . . |ψM 〉}, orto the complementary subset of the remaining states {|ψM+1〉, . . . |ψN 〉} withM < N . To avoid confusion, in this section we denote the a priori proba-bilities of the individual pure states by ηi

j , where j = 1, . . . , N . Note thatfor M = 1 our subset-discrimination problem is also called minimum-errorquantum state filtering, in correspondence to the problem of unambiguousquantum state filtering that has been treated in the respective section ofthis review. The detection operators Π1 and Π2, referring to the two possiblemeasurement outcomes for minimum-error subset-discrimination, are definedin such a way that the quantity 〈ψj |Π1|ψj〉 accounts for the probability toinfer, from performing the measurement, the state of the system to belongto the first subset, if it has been prepared in the state |ψj〉. Obviously, thisinference is correct if j ≤ M . Similarly, the quantity 〈ψj |Π2|ψj〉 is definedas the probability for inferring the state to belong to the second subset. Theoverall error probability reads

PM(N)err = 1−

M∑j=1

ηij〈ψj |Π1|ψj〉+

N∑j=M+1

ηij〈ψj |Π2|ψj〉

, (11.85)

where∑Nj=1 η

ij = 1 and Π1+Π2 = I. In general, with respect to the optimum

measurement strategies, the problem of subset-discrimination is equivalent tothe problem of distinguishing between the two mixed states

ρ1 =1η1

M∑j=1

ηij |ψj〉〈ψj | with η1 =

M∑j=1

ηij , (11.86)

ρ2 =1η2

N∑j=M+1

ηij |ψj〉〈ψj | with η2 =

N∑j=M+1

ηij , (11.87)

provided that the mixed states occur just with the a priori probabilities givenby η1 and η2, respectively [63]. This equivalence also becomes immediatelyobvious from comparing (11.48) and (11.85). Thus the minimum error prob-ability for subset-discrimination can be obtained by applying the Helstromsolution, (11.58), to the problem of discriminating between ρ1 and ρ2.


In our work we adopted another approach that does not entail any increasein the overall calculation effort, but has the advantage of yielding direct infor-mation about the method that realizes the optimum measurement. We firstproved that from the restriction to a two-dimensional Hilbert space it followsthat the optimum detection operator Π1 (or Π2, respectively) can be ex-pressed as the projector onto a particular optimum pure state. By solving theextremal problem that minimizes the expression (11.85) we then determinedthis optimum pure state, as well as the resulting minimum value PM(N)

E .Thus we found that the minimum probability of making an error in distin-guishing to which of the two subsets {|ψ1〉, . . . |ψM 〉} or {|ψM+1〉, . . . |ψN 〉} agiven quantum state belongs is given by [63]

PM(N)E =

12−√R2 + |S|2, (11.88)

where R and S can be expressed as

R =M∑j=1

ηij

(|〈ψ1|ψj〉|2 −

12

)−

N∑j=M+1

ηij

(|〈ψ1|ψj〉|2 −

12

), (11.89)

S =M∑j=1

ηij

〈ψ2|ψj〉〈ψj |ψ1〉 − 〈ψ2|ψ1〉 |〈ψ1|ψj〉|2√1− |〈ψ1|ψ2〉|2

−N∑

j=M+1

ηij

〈ψ2|ψj〉〈ψj |ψ1〉 − 〈ψ2|ψ1〉 |〈ψ1|ψj〉|2√1− |〈ψ1|ψ2〉|2

. (11.90)

In particular, we applied this result to the case of quantum state filtering forthree arbitrary, but linearly dependent states, spanning a two-dimensionalHilbert space. Let us introduce the matrix C by the definition Cij = 〈ψi|ψj〉for i, j = 1, 2, 3. Linear dependence then implies det(C) = 0, or

|〈ψ1|ψ2〉|2 + |〈ψ1|ψ3〉|2 + |〈ψ2|ψ3〉|2 = 1 + 2 Re〈ψ1|ψ2〉〈ψ1|ψ3〉〈ψ1|ψ3〉 .(11.91)

Therefore the minimum error probability obtained in [63] can be written as

P1(3)E =

12−

√√√√14− ηi

2ηi3 (1− |〈ψ2|ψ3〉|2)− ηi

1

3∑j=2

ηij |〈ψ1|ψj〉|2. (11.92)

For ηi3 = 0 the expression reduces to the Helstrom formula (11.59) for dis-

criminating two pure states.As an example, we considered three equally probable symmetric states,

|ψk〉 = cos θ |γ1〉+ ei 2π3 (k−1) sin θ |γ2〉, k = 1, 2, 3 , (11.93)


where |γ1〉 and |γ2〉 denote two orthonormal basis states. For these states alsothe minimum error probability PE for distinguishing them individually can beanalytically expressed, cf. (11.81). We found that the ratio P 1(3)

E (θ)/PE variesbetween 0.56 (for θ ≈ π/12) and 0.5 (for θ = 0 or π/4). Hence the minimum-error probability for distinguishing one state from the set of the two others isonly about half as large as the minimum-error probability for distinguishingall three states separately. For the special example of the equally probabletrine states, given by (11.72), the minimum error probability for quantumstate filtering is obtained to be P 1(3)

E = 1/6. The error-minimizing filteringstrategy consists in performing a projection onto the state |ψ1〉 and guessingthe system to be in this state when the detector clicks, and to be in one of theother states when a projection on the orthogonal state is successful. On theother hand, in the case of the trine states, optimum unambiguous quantumstate filtering would yield the minimum failure probability Q1(3)

F = 1/3, ascan be verified from the formula given in the corresponding chapter of thereview. In this case the measurement corresponds to projecting onto thedirection orthogonal to |ψ1〉, which unambiguously identifies the set of theother states. Obviously the minimized error probability and the optimizedfailure probability for unambiguous discrimination differ just by the factorone half, in agreement with the limit set by (11.66).

Distinguishing a Pure State from a Uniformly Mixed Statein Arbitrary Dimensions

The solution of the minimum-error discrimination problem for two arbitraryquantum states, either pure or mixed, is well known and results in the com-pact Helstrom formula (11.58) for the minimum error probability, PE . How-ever, the explicit analytical evaluation of PE poses severe difficulties when thedimensionality D of the relevant Hilbert space is larger than two. This is dueto the fact that applying the Helstrom formula amounts to calculating theeigenvalues of a D-dimensional matrix. In the following we consider a simpleyet non-trivial example of an error-minimizing state discrimination problemin an arbitrary dimensional Hilbert space that can be solved analytically, and,in addition, might be related to potential applications. Our problem consistsin deciding with minimum error whether a quantum system is prepared eitherin a given pure state or in a given uniformly mixed state [42], i. e. we haveto discriminate between the two quantum states described by

ρ1 = |ψ〉〈ψ|, ρ2 =1d

d∑j=1

|uj〉〈uj |. (11.94)

Here the states |uj〉 are supposed to be mutually orthonormal, i. e. 〈ui|uj〉 =δij . With DS denoting the dimensionality of the physical state space of thequantum system, the relation d ≤ DS has to be fulfilled. We note that in


the special case d = DS the state ρ2 is the maximally mixed state thatdescribes a completely random state of the quantum system, containing noinformation at all. Discriminating between the density operators |ψ〉〈ψ| andρ2 then amounts to deciding whether the state |ψ〉 has been reliably prepared,or whether the preparation has totally failed [44].

To simplify the representation, in the following we restrict ourselves to thespecial case that the a priori probabilities of the two states are η1 = 1/(d+1)for the pure state, and η2 = d/(d+ 1) for the mixed state, respectively [42].This means that in the corresponding quantum state filtering scenario allpossible pure states would have equal a priori probabilities. In order to cal-culate the minimum error probability, using the Helstrom formula (11.58), itis necessary to determine the eigenvalues λ of the operator

Λ =1

d+ 1

d∑j=1

|uj〉〈uj | − |ψ〉〈ψ|

. (11.95)

We found that the eigenvalues are given by [42]

λ1 = − 1d+ 1

√1− ‖ψ‖‖2, (11.96)

λ2 = −λ1, λk =1

d+ 1(k = 3, . . . d+ 1),

where we introduced the notation |ψ‖〉 for the component of |ψ〉 that lies inthe subspace spanned by the states |u1〉, . . . , |ud〉,

‖ψ‖‖2 = 〈ψ‖|ψ‖〉 =d∑j=1

|〈uj |ψ〉|2. (11.97)

When the quantum states to be discriminated are linearly independent, i. e.when ‖ψ‖‖ �= 1, there exists exactly one eigenvalue that is negative, given byλ1. Therefore according to (11.55) the detection operators for performing theminimum-error measurement are given by Π1 = |φ1〉〈φ1| and Π2 = IDS

−Π1,where |φ1〉 is the eigenstate belonging to the negative eigenvalue, λ1. On theother hand, when ρ1 and ρ2 are linearly dependent, i. e. when ‖ψ‖‖ = 1, anegative eigenvalue does not exist. In this case the optimum measurementstrategy is described by the detection operator Π2 = IDS

which means thatthe resulting minimum error probability, PE = 1/(d + 1), is achievable byguessing the system always to be in the state ρ2, without performing anymeasurement at all. The minimum error probability resulting from the aboveeigenvalues reads [42]

PE =1

d+ 1

(1−

√1− ‖ψ‖‖2

). (11.98)


We still mention that the previous considerations can be easily extended tothe case that the pure state and the uniformly mixed state given in (11.94)occur with arbitrary a priori-probabilities η1 and η2 = 1 − η1, respectively.The minimum error probability for distinguishing between them is then givenby [64]

PE =12

[η1 +

η2d−√(

η1 +η2d

)2− 4η1

η2d‖ψ‖‖2

]. (11.99)

Let us now compare the minimum probability of errors, PE , with thesmallest possible failure probability, QF , that can be obtained in a strategyoptimized for unambiguously discriminating between the pure state and theuniformly mixed state. The solution of the latter problem coincides with thesolution to the problem of optimum unambiguous quantum state filtering.Assuming again that η1 = 1/(d + 1), the minimum failure probability isQF = 2 ‖ψ‖‖/(d + 1) [42]. Supposing nonorthogonality of the two states,characterized by 0 < ‖ψ‖‖ ≤ 1, we observe that PE/QF ≤ 1/2, in accordancewith (11.66). When the two states are linearly dependent, i. e. when ‖ψ‖‖ = 1,it follows that PE/QF = 1/2. On the other hand, for nearly orthogonal states,where ‖ψ‖‖ � 1, we find that PE/QF ≈ ‖ψ‖‖/4. Obviously in this case theminimum error probability is drastically smaller than the optimum failureprobability for unambiguous discrimination.

As an application of the minimum-error strategy described above, wediscussed the problem of discriminating between a pure and a mixed two-qubit state [42]. An arbitrary bipartite qubit state, shared among two partiesA (Alice) and B (Bob), can be expressed with the help of the four orthonormalbasis states |00〉, |01〉, |10〉 and |11〉, where |mn〉 stands for |m〉A⊗ |n〉B , with|0〉 and |1〉 denoting any two orthonormal basis states of a single qubit. As aninteresting special case we considered the problem that Alice and Bob wantto decide whether the two-qubit system is either in a given pure state |ψ〉,occuring with the a priori probability η1 = 1/4, or in a uniform mixture ofthe three symmetric states

|u1〉 = |00〉, |u2〉 = |11〉, |u3〉 =12(|01〉+ |10〉). (11.100)

We found that in this case the minimum error probability is given by

PE =14

(1− 1√

2|〈01|ψ〉 − 〈10|ψ〉|

). (11.101)

The same result would hold true if |u1〉 and |u2〉 were replaced by the two sym-metric Bell states (|00〉±|11〉)/

√2. Minimum-error discrimination is achieved

by performing a projection measurement onto the eigenstate |φ1〉 that belongsto the negative eigenvalue λ1 of the operator Λ, cf. (11.95). In general, thiseigenstate will be a superposition of the four two-qubit basis states. The opti-mum measurement strategy therefore requires a correlated measurement that


has to be carried out collectively on the two qubits. On the other hand, ifin our specific example the discrimination would have to performed by localmeasurements on the qubits only, without any communication between Al-ice and Bob, the smallest achievable error probability would be always 1/4,independent of the choice of the state |ψ〉 [42]. We note that the problem weconsidered is of particular interest in the context of quantum state compari-son [30], where one wants to determine whether the states of quantum systemsare identical or not. It has been shown [34] that for comparing two unknownsingle-particle states it is crucial to discriminate the anti-symmetric state ofthe combined two-particle system from the uniform mixture of the mutuallyorthogonal symmetric states. In this context it is interesting that recently anapplication of particle statistics to the problem of minimum-error discrimi-nation has been presented [66]. Discrimination between bipartite states willbe discussed in more detail in a separate chapter.

11.4 Discriminating Multiparticle States

So far we have been considering the situation in which the person discrimi-nating between states is in possession of the entire system that is guaranteedto be in one of the allowed states. However, if our system consists of sub-systems, these can be distributed among different parties, and then theseparties have to determine which state they are sharing by measuring theirsubsystems and communicating among themselves. This adds another layerof complexity to the problem.

The simplest example is to suppose that we have a two-qubit state, andwe give one of the qubits to Alice and the other to Bob. Alice and Bobknow that the state is either |ψ0〉 or |ψ1〉, and by performing local operationsand communicating classically (this is abbreviated as LOCC), they want todetermine which state they have. We shall consider both the case of minimum-error and unambiguous discrimination. The object is to develop a procedurethat Alice and Bob can use to discriminate between the states.

It is possible to immediately obtain some bounds on how successful theseprocedures can be. If both states are equally likely, and both qubits aremeasured together, then we know that the states can be successfully unam-biguously discriminated with a probability of PIDP = 1 − |〈ψ0|ψ1〉| [7]- [9].This clearly represents an upper bound on what can be accomplished usingLOCC. In the case of minimum-error discrimination, the best probability ofcorrectly identifying the state that is obtainable if both states are measuredtogether is

P =12

+12Tr|(η0|ψ0〉〈ψ0| − η1|ψ1〉〈ψ1|)|, (11.102)

where ηj is the a priori probability for |ψj〉, for j = 1, 2. This is again anupper bound to what can be achieved using LOCC. A natural question is


whether these bounds are, in fact, achievable. It was recently shown thatthey are.

In order to see how, let us start with an extreme case; we shall assumethat |ψ0〉 and |ψ1〉 are orthogonal. Walgate, et al. proved that in this casethe states can be distinguished perfectly using only LOCC [67]. They did, infact, much more that this, they showed that two orthogonal states, of anydimension, shared by any number of parties can be perfectly distinguishedby LOCC. Their proof rests on the fact that any two bipartite states can beexpressed in the form

|ψ0〉 =n∑j=0

|j〉A|ξj〉B

|ψ1〉 =n∑j=0

|j〉A|ξ⊥j 〉B , (11.103)

where {|j〉A|j = 1, . . . n} is an orthonormal basis for Alice’s space and thestates |ξj〉B and |ξ⊥

j 〉B , which are not normalized, are orthogonal in Bob’s.Alice measures her state in the |j〉A basis and communicates her result to Bob.If her result was |j0〉, then Bob measures his particle in order to determinewhether it is in the state |ξj0〉B or |ξ⊥

j0〉B , which he can do perfectly since the

states are orthogonal. Note that the measurement that Bob makes dependson the result of Alice’s measurement. If the states are split among more thantwo parties, this procedure can be applied several times. For example, if thereare three parties, Alice, Bob and Charlie, then we initially group Bob andCharlie together so that the state can be considered bipartite. Alice performsher measurement and tells Bob and Charlie the result. They now share one oftwo known, orthogonal states, and they can apply the above procedure againto find out which. The answer will tell them what the original state was.

The case when |ψ0〉 and |ψ1〉 are not orthogonal (and of arbitrary dimen-sion) was investigated by Virmani, et al. [68], and they were able to applythe above decomposition to the problem of minimum-error discrimination.They found a strategy, for arbitrary a priori probabilities, using only LOCCthat achieves the optimal success probability given in (11.102). In addition,they found strong numerical evidence that, when the two states are equallylikely, unambiguous discrimination is possible with a probability of PIDP us-ing LOCC, and they found a class of states for which they could prove thatthis was true. A proof that this is true for all bipartite states was providedby Chen and Yang [69].

The procedure that makes LOCC unambiguous discrimination with a suc-cess probability of PIDP possible is closely related to the one for discriminat-ing orthogonal states. Alice makes a projective measurement on her particlethat gives her no information about whether the state is |ψ0〉 or |ψ1〉, and shethen communicates her result to Bob. Based on what Alice has told him, Bobchooses a measurement to make on his particle. In particular, he applies the


procedure for the optimal unambiguous discrimination of single qubit statesto his particle. However, in this procedure one must know the two states thatone is discriminating between, and it is this information that is provided bythe result of Alice’s measurement.

Together with Mimih we took a somewhat different approach to unam-biguous discrimination of two-qubit states [70]. The motivation was to studybipartite state discrimination schemes that could be used in quantum commu-nication protocols, quantum secret sharing, in particular [71]- [74]. A quan-tum cryptography protocol based on two-state unambiguous discriminationalready existed [38], and this suggested that the discrimination of bipartitestates might find application as well. In secret sharing, Alice and Bob areboth sent information that allows them to decode a message if they act to-gether, but neither party can decode it by themselves. The schemes discussedabove are not well suited for this type of application, because the informationgained by the two parties is not the same. In particular, after the measure-ments have been made (note that Alice must communicate her result to Bobso that he can make his), Alice knows nothing and Bob knows which statehas been sent. We were interested in schemes that are more symmetric.

Our approach was to examine situations in which the classical communi-cation between the parties was limited. One possibility is to allow no classicalcommunication. In that case each party has three possible measurement re-sults, 0 corresponding to |ψ0〉, 1 corresponding to |ψ1〉, and f for failure todistinguish. If |ψ0〉 is sent, then Alice and Bob both measure 0 or both mea-sure f , so that they both know, without communicating, that |ψ0〉 was sentor that the measurement failed. If |ψ1〉 is sent, then they both measure either1 or f . In the case of qubits, we found that the best that can be done is toidentify one of the states and fail the rest of the time, i.e. we never get apositive identification for the second state. The situation improves if we goto qutrits. In that case there are examples of states that can be distinguishedwith the success probability equal to PIDP . One is given by the two states(the states |0〉, |1〉, and |2〉 are an orthonormal basis for the qutrits)

|ψ0〉 =1√2(|00〉+ |22〉)

|ψ1〉 =1√2(|11〉+ |22〉). (11.104)

If Alice and Bob each measure 0, they know they were sent |ψ0〉, if they eachmeasure 1, they know they were sent |ψ1〉, and if they both measure 2 theyfailed.

We also considered the situation in which and Alice and Bob make mea-surements, and later pool their results to determine which state was sent.However, conditional measurements were banned, i.e. situations in which themeasurement made by one party depends on the measurement result of theother were not allowed. Conditional measurements seem to have the problemthat, as has been noted, the information the parties receive is not the same,


and, that if there is a delay between the time the qubits are received andthe time they are measured, then the qubits must be protected against de-coherence until the measurement is made. In the procedures we studied, wefound that it was optimal for each party to make projective measurements.If the two states shared the same Schmidt basis, then these procedures couldsuccessfully distinguish them with a probability of PIDP , but if they did not,the probability of success was, in general, smaller than PIDP .

The discrimination of more than two states shared by two parties has be-gun to be investigated, and presently results only exist for the case in whichall of the states are orthogonal to each other. Ghosh, et al. have shown that itis not possible, in general, to deterministically distinguish either three or fourorthogonal two-qubit states using only local operations and classical commu-nication [75]. A general condition on when orthogonal, bipartite 2× d states(one of the particles is a qubit and the other a qudit), can be distinguishedby LOCC was found recently by Walgate and Hardy [76]. In the case of 2×2states, they found that for three of them to be perfectly distinguishable byLOCC, at least two of them must be product states, and for four, all of themmust be product states. As far as we are aware, the problem of distinguishingthree or more nonorthogonal states shared between two or more parties usingLOCC has not yet been studied although, very recently, Chefles [77] deriveda necessary and sufficient condition for a finite set of states in a finite dimen-sional multiparticle quantum system for LOCC unambiguous discrimination.This suggests that there is much still to be learned about distinguishing mul-tipartite states using local operations and classical communication and, ingeneral, separable quantum operations have to be considered.

11.5 Outlook

Quantum state discrimination is a very rapidly evolving field just like manyother areas of quantum information and quantum computing. We have re-viewed here the two most important - and simplest - state discriminationstrategies, unambiguous discrimination and minimum-error discrimination.The minimum-error strategy for the discrimination of two mixed states hasbeen one of the first problems that was solved exactly. With the recentprogress toward the unambiguous discrimination of mixed states we expectthat in the near future the problem of unambiguous discrimination of twomixed states will be solved completely. Then attention will quite naturallyturn toward the discrimination of more than two states where so far onlyspecial cases were solved completely and partial results were obtained in thegeneral case. A rapidly emerging field with a lot of room for quick progress isthe discrimination of multiparticle states using LOCC only. Applications inthe area of quantum cryptography and probabilistic quantum algorithms willsurely follow but in a field with such a rapidly changing landscape it wouldnot be responsible to predict more than the immediately foreseeable future.


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11 Discrimination of Quantum States · than two states it is diﬃcult to treat mixed states. The...

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